[unix-history] / usr / src / old / libm / libm / log__L.c

/* 
 * Copyright (c) 1985 Regents of the University of California.
 * 
 * Use and reproduction of this software are granted  in  accordance  with
 * the terms and conditions specified in  the  Berkeley  Software  License
 * Agreement (in particular, this entails acknowledgement of the programs'
 * source, and inclusion of this notice) with the additional understanding
 * that  all  recipients  should regard themselves as participants  in  an
 * ongoing  research  project and hence should  feel  obligated  to report
 * their  experiences (good or bad) with these elementary function  codes,
 * using "sendbug 4bsd-bugs@BERKELEY", to the authors.
 */

#ifndef lint
static char sccsid[] = "@(#)log__L.c	1.2 (Berkeley) 8/21/85";
#endif not lint

/* log__L(Z)
 *		LOG(1+X) - 2S			       X
 * RETURN      ---------------  WHERE Z = S*S,  S = ------- , 0 <= Z <= .0294...
 *		      S				     2 + X
 *		     
 * DOUBLE PRECISION (VAX D FORMAT 56 bits or IEEE DOUBLE 53 BITS)
 * KERNEL FUNCTION FOR LOG; TO BE USED IN LOG1P, LOG, AND POW FUNCTIONS
 * CODED IN C BY K.C. NG, 1/19/85; 
 * REVISED BY K.C. Ng, 2/3/85, 4/16/85.
 *
 * Method :
 *	1. Polynomial approximation: let s = x/(2+x). 
 *	   Based on log(1+x) = log(1+s) - log(1-s)
 *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
 *
 *	   (log(1+x) - 2s)/s is computed by
 *
 *	       z*(L1 + z*(L2 + z*(... (L7 + z*L8)...)))
 *
 *	   where z=s*s. (See the listing below for Lk's values.) The 
 *	   coefficients are obtained by a special Remez algorithm. 
 *
 * Accuracy:
 *	Assuming no rounding error, the maximum magnitude of the approximation 
 *	error (absolute) is 2**(-58.49) for IEEE double, and 2**(-63.63)
 *	for VAX D format.
 *
 * 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	/* VAX D format (56 bits) */
/* static double */
/* L1     =  6.6666666666666703212E-1    , Hex  2^  0   *  .AAAAAAAAAAAAC5 */
/* L2     =  3.9999999999970461961E-1    , Hex  2^ -1   *  .CCCCCCCCCC2684 */
/* L3     =  2.8571428579395698188E-1    , Hex  2^ -1   *  .92492492F85782 */
/* L4     =  2.2222221233634724402E-1    , Hex  2^ -2   *  .E38E3839B7AF2C */
/* L5     =  1.8181879517064680057E-1    , Hex  2^ -2   *  .BA2EB4CC39655E */
/* L6     =  1.5382888777946145467E-1    , Hex  2^ -2   *  .9D8551E8C5781D */
/* L7     =  1.3338356561139403517E-1    , Hex  2^ -2   *  .8895B3907FCD92 */
/* L8     =  1.2500000000000000000E-1    , Hex  2^ -2   *  .80000000000000 */
static long        L1x[] = { 0xaaaa402a, 0xaac5aaaa};
static long        L2x[] = { 0xcccc3fcc, 0x2684cccc};
static long        L3x[] = { 0x49243f92, 0x578292f8};
static long        L4x[] = { 0x8e383f63, 0xaf2c39b7};
static long        L5x[] = { 0x2eb43f3a, 0x655ecc39};
static long        L6x[] = { 0x85513f1d, 0x781de8c5};
static long        L7x[] = { 0x95b33f08, 0xcd92907f};
static long        L8x[] = { 0x00003f00, 0x00000000};
#define       L1    (*(double*)L1x)
#define       L2    (*(double*)L2x)
#define       L3    (*(double*)L3x)
#define       L4    (*(double*)L4x)
#define       L5    (*(double*)L5x)
#define       L6    (*(double*)L6x)
#define       L7    (*(double*)L7x)
#define       L8    (*(double*)L8x)
#else	/* IEEE double */
static double
L1     =  6.6666666666667340202E-1    , /*Hex  2^ -1   *  1.5555555555592 */
L2     =  3.9999999999416702146E-1    , /*Hex  2^ -2   *  1.999999997FF24 */
L3     =  2.8571428742008753154E-1    , /*Hex  2^ -2   *  1.24924941E07B4 */
L4     =  2.2222198607186277597E-1    , /*Hex  2^ -3   *  1.C71C52150BEA6 */
L5     =  1.8183562745289935658E-1    , /*Hex  2^ -3   *  1.74663CC94342F */
L6     =  1.5314087275331442206E-1    , /*Hex  2^ -3   *  1.39A1EC014045B */
L7     =  1.4795612545334174692E-1    ; /*Hex  2^ -3   *  1.2F039F0085122 */
#endif

double log__L(z)
double z;
{
#ifdef VAX
    return(z*(L1+z*(L2+z*(L3+z*(L4+z*(L5+z*(L6+z*(L7+z*L8))))))));
#else	/* IEEE double */
    return(z*(L1+z*(L2+z*(L3+z*(L4+z*(L5+z*(L6+z*L7)))))));
#endif
}
Commit	Line	Data
7c0702f7 MAN	1	/*
	2	* Copyright (c) 1985 Regents of the University of California.
	3	*
	4	* Use and reproduction of this software are granted in accordance with
	5	* the terms and conditions specified in the Berkeley Software License
	6	* Agreement (in particular, this entails acknowledgement of the programs'
	7	* source, and inclusion of this notice) with the additional understanding
	8	* that all recipients should regard themselves as participants in an
	9	* ongoing research project and hence should feel obligated to report
	10	* their experiences (good or bad) with these elementary function codes,
	11	* using "sendbug 4bsd-bugs@BERKELEY", to the authors.
	12	*/
	13
	14	#ifndef lint
1c15e888	15	static char sccsid[] = "@(#)log__L.c 1.2 (Berkeley) 8/21/85";
7c0702f7 MAN	16	#endif not lint
	17
	18	/* log__L(Z)
	19	* LOG(1+X) - 2S X
	20	* RETURN --------------- WHERE Z = S*S, S = ------- , 0 <= Z <= .0294...
	21	* S 2 + X
	22	*
	23	* DOUBLE PRECISION (VAX D FORMAT 56 bits or IEEE DOUBLE 53 BITS)
076c6c89	24	* KERNEL FUNCTION FOR LOG; TO BE USED IN LOG1P, LOG, AND POW FUNCTIONS
7c0702f7 MAN	25	* CODED IN C BY K.C. NG, 1/19/85;
	26	* REVISED BY K.C. Ng, 2/3/85, 4/16/85.
	27	*
	28	* Method :
	29	* 1. Polynomial approximation: let s = x/(2+x).
	30	* Based on log(1+x) = log(1+s) - log(1-s)
	31	* = 2s + 2/3 s3 + 2/5 s5 + .....,
	32	*
	33	* (log(1+x) - 2s)/s is computed by
	34	*
	35	* z(L1 + z(L2 + z(... (L7 + zL8)...)))
	36	*
076c6c89	37	* where z=s*s. (See the listing below for Lk's values.) The
7c0702f7 MAN	38	* coefficients are obtained by a special Remez algorithm.
	39	*
	40	* Accuracy:
	41	* Assuming no rounding error, the maximum magnitude of the approximation
	42	* error (absolute) is 2(-58.49) for IEEE double, and 2(-63.63)
	43	* for VAX D format.
	44	*
	45	* Constants:
	46	* The hexadecimal values are the intended ones for the following constants.
	47	* The decimal values may be used, provided that the compiler will convert
	48	* from decimal to binary accurately enough to produce the hexadecimal values
	49	* shown.
	50	*/
	51
	52	#ifdef VAX /* VAX D format (56 bits) */
	53	/* static double */
	54	/* L1 = 6.6666666666666703212E-1 , Hex 2^ 0 * .AAAAAAAAAAAAC5 */
	55	/* L2 = 3.9999999999970461961E-1 , Hex 2^ -1 * .CCCCCCCCCC2684 */
	56	/* L3 = 2.8571428579395698188E-1 , Hex 2^ -1 * .92492492F85782 */
	57	/* L4 = 2.2222221233634724402E-1 , Hex 2^ -2 * .E38E3839B7AF2C */
	58	/* L5 = 1.8181879517064680057E-1 , Hex 2^ -2 * .BA2EB4CC39655E */
	59	/* L6 = 1.5382888777946145467E-1 , Hex 2^ -2 * .9D8551E8C5781D */
	60	/* L7 = 1.3338356561139403517E-1 , Hex 2^ -2 * .8895B3907FCD92 */
	61	/* L8 = 1.2500000000000000000E-1 , Hex 2^ -2 * .80000000000000 */
	62	static long L1x[] = { 0xaaaa402a, 0xaac5aaaa};
	63	static long L2x[] = { 0xcccc3fcc, 0x2684cccc};
	64	static long L3x[] = { 0x49243f92, 0x578292f8};
	65	static long L4x[] = { 0x8e383f63, 0xaf2c39b7};
	66	static long L5x[] = { 0x2eb43f3a, 0x655ecc39};
	67	static long L6x[] = { 0x85513f1d, 0x781de8c5};
	68	static long L7x[] = { 0x95b33f08, 0xcd92907f};
	69	static long L8x[] = { 0x00003f00, 0x00000000};
	70	#define L1 ((double)L1x)
	71	#define L2 ((double)L2x)
	72	#define L3 ((double)L3x)
	73	#define L4 ((double)L4x)
	74	#define L5 ((double)L5x)
	75	#define L6 ((double)L6x)
	76	#define L7 ((double)L7x)
	77	#define L8 ((double)L8x)
076c6c89	78	#else /* IEEE double */
7c0702f7 MAN	79	static double
	80	L1 = 6.6666666666667340202E-1 , /Hex 2^ -1 1.5555555555592 */
	81	L2 = 3.9999999999416702146E-1 , /Hex 2^ -2 1.999999997FF24 */
	82	L3 = 2.8571428742008753154E-1 , /Hex 2^ -2 1.24924941E07B4 */
	83	L4 = 2.2222198607186277597E-1 , /Hex 2^ -3 1.C71C52150BEA6 */
	84	L5 = 1.8183562745289935658E-1 , /Hex 2^ -3 1.74663CC94342F */
	85	L6 = 1.5314087275331442206E-1 , /Hex 2^ -3 1.39A1EC014045B */
	86	L7 = 1.4795612545334174692E-1 ; /Hex 2^ -3 1.2F039F0085122 */
	87	#endif
	88
	89	double log__L(z)
	90	double z;
	91	{
	92	#ifdef VAX
	93	return(z(L1+z(L2+z(L3+z(L4+z(L5+z(L6+z(L7+zL8))))))));
076c6c89	94	#else /* IEEE double */
7c0702f7 MAN	95	return(z(L1+z(L2+z(L3+z(L4+z(L5+z(L6+z*L7)))))));
	96	#endif
	97	}