trig.c

<B>Digital的Unix操作系统VAX 4.2源码</B>

#ifndef lint
static char	*sccsid = " @(#)trig.c	4.1	(ULTRIX)	7/17/90";
#endif lint

/************************************************************************
 *									*
 *			Copyright (c) 1986 by				*
 *		Digital Equipment Corporation, Maynard, MA		*
 *			All rights reserved.				*
 *									*
 *   This software is furnished under a license and may be used and	*
 *   copied  only  in accordance with the terms of such license and	*
 *   with the  inclusion  of  the  above  copyright  notice.   This	*
 *   software  or  any  other copies thereof may not be provided or	*
 *   otherwise made available to any other person.  No title to and	*
 *   ownership of the software is hereby transferred.			*
 *									*
 *   This software is  derived  from  software  received  from  the	*
 *   University    of   California,   Berkeley,   and   from   Bell	*
 *   Laboratories.  Use, duplication, or disclosure is  subject  to	*
 *   restrictions  under  license  agreements  with  University  of	*
 *   California and with AT&T.						*
 *									*
 *   The information in this software is subject to change  without	*
 *   notice  and should not be construed as a commitment by Digital	*
 *   Equipment Corporation.						*
 *									*
 *   Digital assumes no responsibility for the use  or  reliability	*
 *   of its software on equipment which is not supplied by Digital.	*
 *									*
 ************************************************************************/

/************************************************************************
*
*			Modification History
*
*		David Metsky		13-Jan-86
*
* 001	Added from BSD 4.3 version as part of upgrade
*
*	Based on:	trig.c		1.2		8/22/85
*
*************************************************************************/

/* SIN(X), COS(X), TAN(X)
 * RETURN THE SINE, COSINE, AND TANGENT OF X RESPECTIVELY
 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
 * CODED IN C BY K.C. NG, 1/8/85; 
 * REVISED BY W. Kahan and K.C. NG, 8/17/85.
 *
 * Required system supported functions:
 *      copysign(x,y)
 *      finite(x)
 *      drem(x,p)
 *
 * Static kernel functions:
 *      sin__S(z)       ....sin__S(x*x) return (sin(x)-x)/x
 *      cos__C(z)       ....cos__C(x*x) return cos(x)-1-x*x/2
 *
 * Method.
 *      Let S and C denote the polynomial approximations to sin and cos 
 *      respectively on [-PI/4, +PI/4].
 *
 *      SIN and COS:
 *      1. Reduce the argument into [-PI , +PI] by the remainder function.  
 *      2. For x in (-PI,+PI), there are three cases:
 *			case 1:	|x| < PI/4
 *			case 2:	PI/4 <= |x| < 3PI/4
 *			case 3:	3PI/4 <= |x|.
 *	   SIN and COS of x are computed by:
 *
 *                   sin(x)      cos(x)       remark
 *     ----------------------------------------------------------
 *        case 1     S(x)         C(x)       
 *        case 2 sign(x)*C(y)     S(y)      y=PI/2-|x|
 *        case 3     S(y)        -C(y)      y=sign(x)*(PI-|x|)
 *     ----------------------------------------------------------
 *
 *      TAN:
 *      1. Reduce the argument into [-PI/2 , +PI/2] by the remainder function.  
 *      2. For x in (-PI/2,+PI/2), there are two cases:
 *			case 1:	|x| < PI/4
 *			case 2:	PI/4 <= |x| < PI/2
 *         TAN of x is computed by:
 *
 *                   tan (x)            remark
 *     ----------------------------------------------------------
 *        case 1     S(x)/C(x)
 *        case 2     C(y)/S(y)     y=sign(x)*(PI/2-|x|)
 *     ----------------------------------------------------------
 *
 *   Notes:
 *      1. S(y) and C(y) were computed by:
 *              S(y) = y+y*sin__S(y*y) 
 *              C(y) = 1-(y*y/2-cos__C(x*x))          ... if y*y/2 <  thresh,
 *                   = 0.5-((y*y/2-0.5)-cos__C(x*x))  ... if y*y/2 >= thresh.
 *         where
 *              thresh = 0.5*(acos(3/4)**2)
 *
 *      2. For better accuracy, we use the following formula for S/C for tan
 *         (k=0): let ss=sin__S(y*y), and cc=cos__C(y*y), then
 *
 *                            y+y*ss             (y*y/2-cc)+ss
 *             S(y)/C(y)   = -------- = y + y * ---------------.
 *                               C                     C 
 *
 *
 * Special cases:
 *      Let trig be any of sin, cos, or tan.
 *      trig(+-INF)  is NaN, with signals;
 *      trig(NaN)    is that NaN;
 *      trig(n*PI/2) is exact for any integer n, provided n*PI is 
 *      representable; otherwise, trig(x) is inexact. 
 *
 * Accuracy:
 *      trig(x) returns the exact trig(x*pi/PI) nearly rounded, where
 *
 *      Decimal:
 *              pi = 3.141592653589793 23846264338327 ..... 
 *    53 bits   PI = 3.141592653589793 115997963 ..... ,
 *    56 bits   PI = 3.141592653589793 227020265 ..... ,  
 *
 *      Hexadecimal:
 *              pi = 3.243F6A8885A308D313198A2E....
 *    53 bits   PI = 3.243F6A8885A30  =  2 * 1.921FB54442D18    error=.276ulps
 *    56 bits   PI = 3.243F6A8885A308 =  4 * .C90FDAA22168C2    error=.206ulps
 *
 *      In a test run with 1,024,000 random arguments on a VAX, the maximum
 *      observed errors (compared with the exact trig(x*pi/PI)) were
 *                      tan(x) : 2.09 ulps (around 4.716340404662354)
 *                      sin(x) : .861 ulps
 *                      cos(x) : .857 ulps
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following constants.
 * The decimal values may be used, provided that the compiler will convert
 * from decimal to binary accurately enough to produce the hexadecimal values
 * shown.
 */

#ifdef VAX
/*thresh =  2.6117239648121182150E-1    , Hex  2^ -1   *  .85B8636B026EA0 */
/*PIo4   =  7.8539816339744830676E-1    , Hex  2^  0   *  .C90FDAA22168C2 */
/*PIo2   =  1.5707963267948966135E0     , Hex  2^  1   *  .C90FDAA22168C2 */
/*PI3o4  =  2.3561944901923449203E0     , Hex  2^  2   *  .96CBE3F9990E92 */
/*PI     =  3.1415926535897932270E0     , Hex  2^  2   *  .C90FDAA22168C2 */
/*PI2    =  6.2831853071795864540E0     ; Hex  2^  3   *  .C90FDAA22168C2 */
static long    threshx[] = { 0xb8633f85, 0x6ea06b02};
#define   thresh    (*(double*)threshx)
static long      PIo4x[] = { 0x0fda4049, 0x68c2a221};
#define     PIo4    (*(double*)PIo4x)
static long      PIo2x[] = { 0x0fda40c9, 0x68c2a221};
#define     PIo2    (*(double*)PIo2x)
static long      PI3o4x[] = { 0xcbe34116, 0x0e92f999};
#define     PI3o4    (*(double*)PI3o4x)
static long        PIx[] = { 0x0fda4149, 0x68c2a221};
#define       PI    (*(double*)PIx)
static long       PI2x[] = { 0x0fda41c9, 0x68c2a221};
#define      PI2    (*(double*)PI2x)
#else   /* IEEE double  */
static double
thresh =  2.6117239648121182150E-1    , /*Hex  2^ -2   *  1.0B70C6D604DD4 */
PIo4   =  7.8539816339744827900E-1    , /*Hex  2^ -1   *  1.921FB54442D18 */
PIo2   =  1.5707963267948965580E0     , /*Hex  2^  0   *  1.921FB54442D18 */
PI3o4  =  2.3561944901923448370E0     , /*Hex  2^  1   *  1.2D97C7F3321D2 */
PI     =  3.1415926535897931160E0     , /*Hex  2^  1   *  1.921FB54442D18 */
PI2    =  6.2831853071795862320E0     ; /*Hex  2^  2   *  1.921FB54442D18 */
#endif
static double zero=0, one=1, negone= -1, half=1.0/2.0, 
	      small=1E-10, /* 1+small**2==1; better values for small:
					small = 1.5E-9 for VAX D
					      = 1.2E-8 for IEEE Double
					      = 2.8E-10 for IEEE Extended */
	      big=1E20;    /* big = 1/(small**2) */

double tan(x) 
double x;
{
        double copysign(),drem(),cos__C(),sin__S(),a,z,ss,cc,c;
        int finite(),k;

        /* tan(NaN) and tan(INF) must be NaN */
            if(!finite(x))  return(x-x);
        x=drem(x,PI);        /* reduce x into [-PI/2, PI/2] */
        a=copysign(x,one);   /* ... = abs(x) */
	if ( a >= PIo4 ) {k=1; x = copysign( PIo2 - a , x ); }
	   else { k=0; if(a < small ) { big + a; return(x); }}

        z  = x*x;
        cc = cos__C(z);
        ss = sin__S(z);
	z  = z*half ;		/* Next get c = cos(x) accurately */
	c  = (z >= thresh )? half-((z-half)-cc) : one-(z-cc);
	if (k==0) return ( x + (x*(z-(cc-ss)))/c );  /* sin/cos */
	return( c/(x+x*ss) );	/*                  ... cos/sin */


}
double sin(x)
double x;
{
        double copysign(),drem(),sin__S(),cos__C(),a,c,z;
        int finite();

        /* sin(NaN) and sin(INF) must be NaN */
            if(!finite(x))  return(x-x);
	x=drem(x,PI2);         /*    reduce x into [-PI, PI] */
        a=copysign(x,one);
	if( a >= PIo4 ) {
	     if( a >= PI3o4 )   /* 	.. in [3PI/4,  PI ]  */