[unix-history] / usr / src / lib / libm / common_source / log1p.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[] =
"@(#)log1p.c	1.3 (Berkeley) 8/21/85; 1.3 (ucb.elefunt) %G%";
#endif not lint

/* LOG1P(x) 
 * RETURN THE LOGARITHM OF 1+x
 * DOUBLE PRECISION (VAX D FORMAT 56 bits, IEEE DOUBLE 53 BITS)
 * CODED IN C BY K.C. NG, 1/19/85; 
 * REVISED BY K.C. NG on 2/6/85, 3/7/85, 3/24/85, 4/16/85.
 * 
 * Required system supported functions:
 *	scalb(x,n) 
 *	copysign(x,y)
 *	logb(x)	
 *	finite(x)
 *
 * Required kernel function:
 *	log__L(z)
 *
 * Method :
 *	1. Argument Reduction: find k and f such that 
 *			1+x  = 2^k * (1+f), 
 *	   where  sqrt(2)/2 < 1+f < sqrt(2) .
 *
 *	2. Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
 *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
 *	   log(1+f) is computed by
 *
 *	     		log(1+f) = 2s + s*log__L(s*s)
 *	   where
 *		log__L(z) = z*(L1 + z*(L2 + z*(... (L6 + z*L7)...)))
 *
 *	   See log__L() for the values of the coefficients.
 *
 *	3. Finally,  log(1+x) = k*ln2 + log(1+f).  
 *
 *	Remarks 1. In step 3 n*ln2 will be stored in two floating point numbers
 *		   n*ln2hi + n*ln2lo, where ln2hi is chosen such that the last 
 *		   20 bits (for VAX D format), or the last 21 bits ( for IEEE 
 *		   double) is 0. This ensures n*ln2hi is exactly representable.
 *		2. In step 1, f may not be representable. A correction term c
 *	 	   for f is computed. It follows that the correction term for
 *		   f - t (the leading term of log(1+f) in step 2) is c-c*x. We
 *		   add this correction term to n*ln2lo to attenuate the error.
 *
 *
 * Special cases:
 *	log1p(x) is NaN with signal if x < -1; log1p(NaN) is NaN with no signal;
 *	log1p(INF) is +INF; log1p(-1) is -INF with signal;
 *	only log1p(0)=0 is exact for finite argument.
 *
 * Accuracy:
 *	log1p(x) returns the exact log(1+x) nearly rounded. In a test run 
 *	with 1,536,000 random arguments on a VAX, the maximum observed
 *	error was .846 ulps (units in the last place).
 *
 * 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 */
#include <errno.h>

/* static double */
/* ln2hi  =  6.9314718055829871446E-1    , Hex  2^  0   *  .B17217F7D00000 */
/* ln2lo  =  1.6465949582897081279E-12   , Hex  2^-39   *  .E7BCD5E4F1D9CC */
/* sqrt2  =  1.4142135623730950622E0     ; Hex  2^  1   *  .B504F333F9DE65 */
static long     ln2hix[] = { 0x72174031, 0x0000f7d0};
static long     ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1};
static long     sqrt2x[] = { 0x04f340b5, 0xde6533f9};
#define    ln2hi    (*(double*)ln2hix)
#define    ln2lo    (*(double*)ln2lox)
#define    sqrt2    (*(double*)sqrt2x)
#else	/* IEEE double */
static double
ln2hi  =  6.9314718036912381649E-1    , /*Hex  2^ -1   *  1.62E42FEE00000 */
ln2lo  =  1.9082149292705877000E-10   , /*Hex  2^-33   *  1.A39EF35793C76 */
sqrt2  =  1.4142135623730951455E0     ; /*Hex  2^  0   *  1.6A09E667F3BCD */
#endif

double log1p(x)
double x;
{
	static double zero=0.0, negone= -1.0, one=1.0, 
		      half=1.0/2.0, small=1.0E-20;   /* 1+small == 1 */
	double logb(),copysign(),scalb(),log__L(),z,s,t,c;
	int k,finite();

#ifndef VAX
	if(x!=x) return(x);	/* x is NaN */
#endif

	if(finite(x)) {
	   if( x > negone ) {

	   /* argument reduction */
	      if(copysign(x,one)<small) return(x);
	      k=logb(one+x); z=scalb(x,-k); t=scalb(one,-k);
	      if(z+t >= sqrt2 ) 
		  { k += 1 ; z *= half; t *= half; }
	      t += negone; x = z + t;
	      c = (t-x)+z ;		/* correction term for x */

 	   /* compute log(1+x)  */
              s = x/(2+x); t = x*x*half;
	      c += (k*ln2lo-c*x);
	      z = c+s*(t+log__L(s*s));
	      x += (z - t) ;

	      return(k*ln2hi+x);
	   }
	/* end of if (x > negone) */

	    else {
#ifdef VAX
		extern double infnan();
		if ( x == negone )
		    return (infnan(-ERANGE));	/* -INF */
		else
		    return (infnan(EDOM));	/* NaN */
#else	/* IEEE double */
		/* x = -1, return -INF with signal */
		if ( x == negone ) return( negone/zero );

		/* negative argument for log, return NaN with signal */
	        else return ( zero / zero );
#endif
	    }
	}
    /* end of if (finite(x)) */

    /* log(-INF) is NaN */
	else if(x<0) 
	     return(zero/zero);

    /* log(+INF) is INF */
	else return(x);      
}
Commit	Line	Data
ed7e5132 ZAL	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
a62df508	15	static char sccsid[] =
62b65e15	16	"@(#)log1p.c 1.3 (Berkeley) 8/21/85; 1.3 (ucb.elefunt) %G%";
ed7e5132 ZAL	17	#endif not lint
	18
	19	/* LOG1P(x)
	20	* RETURN THE LOGARITHM OF 1+x
	21	* DOUBLE PRECISION (VAX D FORMAT 56 bits, IEEE DOUBLE 53 BITS)
	22	* CODED IN C BY K.C. NG, 1/19/85;
	23	* REVISED BY K.C. NG on 2/6/85, 3/7/85, 3/24/85, 4/16/85.
	24	*
	25	* Required system supported functions:
	26	* scalb(x,n)
	27	* copysign(x,y)
	28	* logb(x)
	29	* finite(x)
	30	*
	31	* Required kernel function:
	32	* log__L(z)
	33	*
	34	* Method :
	35	* 1. Argument Reduction: find k and f such that
	36	* 1+x = 2^k * (1+f),
	37	* where sqrt(2)/2 < 1+f < sqrt(2) .
	38	*
	39	* 2. Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
	40	* = 2s + 2/3 s3 + 2/5 s5 + .....,
	41	* log(1+f) is computed by
	42	*
	43	* log(1+f) = 2s + slog__L(ss)
	44	* where
	45	* log__L(z) = z(L1 + z(L2 + z(... (L6 + zL7)...)))
	46	*
	47	* See log__L() for the values of the coefficients.
	48	*
	49	* 3. Finally, log(1+x) = k*ln2 + log(1+f).
	50	*
	51	* Remarks 1. In step 3 n*ln2 will be stored in two floating point numbers
	52	* nln2hi + nln2lo, where ln2hi is chosen such that the last
	53	* 20 bits (for VAX D format), or the last 21 bits ( for IEEE
	54	* double) is 0. This ensures n*ln2hi is exactly representable.
	55	* 2. In step 1, f may not be representable. A correction term c
	56	* for f is computed. It follows that the correction term for
	57	* f - t (the leading term of log(1+f) in step 2) is c-c*x. We
	58	* add this correction term to n*ln2lo to attenuate the error.
	59	*
	60	*
	61	* Special cases:
	62	* log1p(x) is NaN with signal if x < -1; log1p(NaN) is NaN with no signal;
	63	* log1p(INF) is +INF; log1p(-1) is -INF with signal;
	64	* only log1p(0)=0 is exact for finite argument.
	65	*
	66	* Accuracy:
	67	* log1p(x) returns the exact log(1+x) nearly rounded. In a test run
	68	* with 1,536,000 random arguments on a VAX, the maximum observed
	69	* error was .846 ulps (units in the last place).
	70	*
	71	* Constants:
	72	* The hexadecimal values are the intended ones for the following constants.
	73	* The decimal values may be used, provided that the compiler will convert
	74	* from decimal to binary accurately enough to produce the hexadecimal values
	75	* shown.
	76	*/
	77
	78	#ifdef VAX /* VAX D format */
	79	#include <errno.h>
	80
62b65e15	81	/* static double */
ed7e5132 ZAL	82	/* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */
	83	/* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */
	84	/* sqrt2 = 1.4142135623730950622E0 ; Hex 2^ 1 * .B504F333F9DE65 */
	85	static long ln2hix[] = { 0x72174031, 0x0000f7d0};
	86	static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1};
	87	static long sqrt2x[] = { 0x04f340b5, 0xde6533f9};
	88	#define ln2hi ((double)ln2hix)
	89	#define ln2lo ((double)ln2lox)
	90	#define sqrt2 ((double)sqrt2x)
	91	#else /* IEEE double */
62b65e15	92	static double
ed7e5132 ZAL	93	ln2hi = 6.9314718036912381649E-1 , /Hex 2^ -1 1.62E42FEE00000 */
	94	ln2lo = 1.9082149292705877000E-10 , /Hex 2^-33 1.A39EF35793C76 */
	95	sqrt2 = 1.4142135623730951455E0 ; /Hex 2^ 0 1.6A09E667F3BCD */
	96	#endif
	97
	98	double log1p(x)
	99	double x;
	100	{
	101	static double zero=0.0, negone= -1.0, one=1.0,
	102	half=1.0/2.0, small=1.0E-20; /* 1+small == 1 */
	103	double logb(),copysign(),scalb(),log__L(),z,s,t,c;
	104	int k,finite();
	105
	106	#ifndef VAX
	107	if(x!=x) return(x); /* x is NaN */
	108	#endif
	109
	110	if(finite(x)) {
	111	if( x > negone ) {
	112
	113	/* argument reduction */
	114	if(copysign(x,one)<small) return(x);
	115	k=logb(one+x); z=scalb(x,-k); t=scalb(one,-k);
	116	if(z+t >= sqrt2 )
	117	{ k += 1 ; z = half; t = half; }
	118	t += negone; x = z + t;
	119	c = (t-x)+z ; /* correction term for x */
	120
	121	/* compute log(1+x) */
	122	s = x/(2+x); t = xxhalf;
	123	c += (kln2lo-cx);
	124	z = c+s(t+log__L(ss));
	125	x += (z - t) ;
	126
	127	return(k*ln2hi+x);
	128	}
	129	/* end of if (x > negone) */
	130
	131	else {
	132	#ifdef VAX
	133	extern double infnan();
	134	if ( x == negone )
	135	return (infnan(-ERANGE)); /* -INF */
	136	else
	137	return (infnan(EDOM)); /* NaN */
	138	#else /* IEEE double */
	139	/* x = -1, return -INF with signal */
	140	if ( x == negone ) return( negone/zero );
	141
	142	/* negative argument for log, return NaN with signal */
	143	else return ( zero / zero );
	144	#endif
	145	}
	146	}
	147	/* end of if (finite(x)) */
	148
	149	/* log(-INF) is NaN */
	150	else if(x<0)
	151	return(zero/zero);
	152
	153	/* log(+INF) is INF */
	154	else return(x);
	155	}