[unix-history] / usr / src / lib / libm / common_source / expm1.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[] =
"@(#)expm1.c	1.2 (Berkeley) 8/21/85; 1.5 (ucb.elefunt) %G%";
#endif not lint

/* EXPM1(X)
 * RETURN THE EXPONENTIAL OF X MINUS ONE
 * DOUBLE PRECISION (IEEE 53 BITS, VAX D FORMAT 56 BITS)
 * CODED IN C BY K.C. NG, 1/19/85; 
 * REVISED BY K.C. NG on 2/6/85, 3/7/85, 3/21/85, 4/16/85.
 *
 * Required system supported functions:
 *	scalb(x,n)	
 *	copysign(x,y)	
 *	finite(x)
 *
 * Kernel function:
 *	exp__E(x,c)
 *
 * Method:
 *	1. Argument Reduction: given the input x, find r and integer k such 
 *	   that
 *	                   x = k*ln2 + r,  |r| <= 0.5*ln2 .  
 *	   r will be represented as r := z+c for better accuracy.
 *
 *	2. Compute EXPM1(r)=exp(r)-1 by 
 *
 *			EXPM1(r=z+c) := z + exp__E(z,c)
 *
 *	3. EXPM1(x) =  2^k * ( EXPM1(r) + 1-2^-k ).
 *
 * 	Remarks: 
 *	   1. When k=1 and z < -0.25, we use the following formula for
 *	      better accuracy:
 *			EXPM1(x) = 2 * ( (z+0.5) + exp__E(z,c) )
 *	   2. To avoid rounding error in 1-2^-k where k is large, we use
 *			EXPM1(x) = 2^k * { [z+(exp__E(z,c)-2^-k )] + 1 }
 *	      when k>56. 
 *
 * Special cases:
 *	EXPM1(INF) is INF, EXPM1(NaN) is NaN;
 *	EXPM1(-INF)= -1;
 *	for finite argument, only EXPM1(0)=0 is exact.
 *
 * Accuracy:
 *	EXPM1(x) returns the exact (exp(x)-1) nearly rounded. In a test run with
 *	1,166,000 random arguments on a VAX, the maximum observed error was
 *	.872 ulps (units of 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.
 */

#if (defined(VAX)||defined(TAHOE))	/* VAX D format */
#ifdef VAX
#define _0x(A,B)	0x/**/A/**/B
#else	/* VAX */
#define _0x(A,B)	0x/**/B/**/A
#endif	/* VAX */
/* static double */
/* ln2hi  =  6.9314718055829871446E-1    , Hex  2^  0   *  .B17217F7D00000 */
/* ln2lo  =  1.6465949582897081279E-12   , Hex  2^-39   *  .E7BCD5E4F1D9CC */
/* lnhuge =  9.4961163736712506989E1     , Hex  2^  7   *  .BDEC1DA73E9010 */
/* invln2 =  1.4426950408889634148E0     ; Hex  2^  1   *  .B8AA3B295C17F1 */
static long     ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)};
static long     ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)};
static long    lnhugex[] = { _0x(ec1d,43bd), _0x(9010,a73e)};
static long    invln2x[] = { _0x(aa3b,40b8), _0x(17f1,295c)};
#define    ln2hi    (*(double*)ln2hix)
#define    ln2lo    (*(double*)ln2lox)
#define   lnhuge    (*(double*)lnhugex)
#define   invln2    (*(double*)invln2x)
#else	/* IEEE double */
static double
ln2hi  =  6.9314718036912381649E-1    , /*Hex  2^ -1   *  1.62E42FEE00000 */
ln2lo  =  1.9082149292705877000E-10   , /*Hex  2^-33   *  1.A39EF35793C76 */
lnhuge =  7.1602103751842355450E2     , /*Hex  2^  9   *  1.6602B15B7ECF2 */
invln2 =  1.4426950408889633870E0     ; /*Hex  2^  0   *  1.71547652B82FE */
#endif

double expm1(x)
double x;
{
	static double one=1.0, half=1.0/2.0; 
	double scalb(), copysign(), exp__E(), z,hi,lo,c;
	int k,finite();
#if (defined(VAX)||defined(TAHOE))
	static prec=56;
#else	/* IEEE double */
	static prec=53;
#endif
#if (!defined(VAX)&&!defined(TAHOE))
	if(x!=x) return(x);	/* x is NaN */
#endif

	if( x <= lnhuge ) {
		if( x >= -40.0 ) {

		    /* argument reduction : x - k*ln2 */
			k= invln2 *x+copysign(0.5,x);	/* k=NINT(x/ln2) */
			hi=x-k*ln2hi ; 
			z=hi-(lo=k*ln2lo);
			c=(hi-z)-lo;

			if(k==0) return(z+exp__E(z,c));
			if(k==1)
			    if(z< -0.25) 
				{x=z+half;x +=exp__E(z,c); return(x+x);}
			    else
				{z+=exp__E(z,c); x=half+z; return(x+x);}
		    /* end of k=1 */

			else {
			    if(k<=prec)
			      { x=one-scalb(one,-k); z += exp__E(z,c);}
			    else if(k<100)
			      { x = exp__E(z,c)-scalb(one,-k); x+=z; z=one;}
			    else 
			      { x = exp__E(z,c)+z; z=one;}

			    return (scalb(x+z,k));  
			}
		}
		/* end of x > lnunfl */

		else 
		     /* expm1(-big#) rounded to -1 (inexact) */
		     if(finite(x))  
			 { ln2hi+ln2lo; return(-one);}

		     /* expm1(-INF) is -1 */
		     else return(-one);
	}
	/* end of x < lnhuge */

	else 
	/*  expm1(INF) is INF, expm1(+big#) overflows to INF */
	    return( finite(x) ?  scalb(one,5000) : x);
}
Commit	Line	Data
e5526aba 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[] =
0e01cbea	16	"@(#)expm1.c 1.2 (Berkeley) 8/21/85; 1.5 (ucb.elefunt) %G%";
e5526aba ZAL	17	#endif not lint
	18
	19	/* EXPM1(X)
	20	* RETURN THE EXPONENTIAL OF X MINUS ONE
	21	* DOUBLE PRECISION (IEEE 53 BITS, VAX D FORMAT 56 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, 3/21/85, 4/16/85.
	24	*
	25	* Required system supported functions:
	26	* scalb(x,n)
	27	* copysign(x,y)
	28	* finite(x)
	29	*
	30	* Kernel function:
	31	* exp__E(x,c)
	32	*
	33	* Method:
	34	* 1. Argument Reduction: given the input x, find r and integer k such
	35	* that
	36	* x = kln2 + r, \|r\| <= 0.5ln2 .
	37	* r will be represented as r := z+c for better accuracy.
	38	*
	39	* 2. Compute EXPM1(r)=exp(r)-1 by
	40	*
	41	* EXPM1(r=z+c) := z + exp__E(z,c)
	42	*
	43	* 3. EXPM1(x) = 2^k * ( EXPM1(r) + 1-2^-k ).
	44	*
	45	* Remarks:
	46	* 1. When k=1 and z < -0.25, we use the following formula for
	47	* better accuracy:
	48	* EXPM1(x) = 2 * ( (z+0.5) + exp__E(z,c) )
	49	* 2. To avoid rounding error in 1-2^-k where k is large, we use
	50	* EXPM1(x) = 2^k * { [z+(exp__E(z,c)-2^-k )] + 1 }
	51	* when k>56.
	52	*
	53	* Special cases:
	54	* EXPM1(INF) is INF, EXPM1(NaN) is NaN;
	55	* EXPM1(-INF)= -1;
	56	* for finite argument, only EXPM1(0)=0 is exact.
	57	*
	58	* Accuracy:
	59	* EXPM1(x) returns the exact (exp(x)-1) nearly rounded. In a test run with
	60	* 1,166,000 random arguments on a VAX, the maximum observed error was
	61	* .872 ulps (units of the last place).
	62	*
	63	* Constants:
	64	* The hexadecimal values are the intended ones for the following constants.
	65	* The decimal values may be used, provided that the compiler will convert
	66	* from decimal to binary accurately enough to produce the hexadecimal values
	67	* shown.
	68	*/
	69
e0085737	70	#if (defined(VAX)\|\|defined(TAHOE)) /* VAX D format */
0e01cbea ZAL	71	#ifdef VAX
	72	#define _0x(A,B) 0x//A//B
	73	#else /* VAX */
	74	#define _0x(A,B) 0x//B//A
	75	#endif /* VAX */
62b65e15	76	/* static double */
e5526aba ZAL	77	/* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */
	78	/* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */
	79	/* lnhuge = 9.4961163736712506989E1 , Hex 2^ 7 * .BDEC1DA73E9010 */
	80	/* invln2 = 1.4426950408889634148E0 ; Hex 2^ 1 * .B8AA3B295C17F1 */
0e01cbea ZAL	81	static long ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)};
	82	static long ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)};
	83	static long lnhugex[] = { _0x(ec1d,43bd), _0x(9010,a73e)};
	84	static long invln2x[] = { _0x(aa3b,40b8), _0x(17f1,295c)};
e5526aba ZAL	85	#define ln2hi ((double)ln2hix)
	86	#define ln2lo ((double)ln2lox)
	87	#define lnhuge ((double)lnhugex)
	88	#define invln2 ((double)invln2x)
	89	#else /* IEEE double */
62b65e15	90	static double
e5526aba ZAL	91	ln2hi = 6.9314718036912381649E-1 , /Hex 2^ -1 1.62E42FEE00000 */
	92	ln2lo = 1.9082149292705877000E-10 , /Hex 2^-33 1.A39EF35793C76 */
	93	lnhuge = 7.1602103751842355450E2 , /Hex 2^ 9 1.6602B15B7ECF2 */
	94	invln2 = 1.4426950408889633870E0 ; /Hex 2^ 0 1.71547652B82FE */
	95	#endif
	96
	97	double expm1(x)
	98	double x;
	99	{
62b65e15	100	static double one=1.0, half=1.0/2.0;
e5526aba ZAL	101	double scalb(), copysign(), exp__E(), z,hi,lo,c;
e5526aba ZAL	102	int k,finite();
e0085737	103	#if (defined(VAX)\|\|defined(TAHOE))
e5526aba ZAL	104	static prec=56;
	105	#else /* IEEE double */
	106	static prec=53;
	107	#endif
e0085737	108	#if (!defined(VAX)&&!defined(TAHOE))
e5526aba ZAL	109	if(x!=x) return(x); /* x is NaN */
	110	#endif
	111
	112	if( x <= lnhuge ) {
	113	if( x >= -40.0 ) {
	114
	115	/* argument reduction : x - kln2 /
	116	k= invln2 x+copysign(0.5,x); / k=NINT(x/ln2) */
	117	hi=x-k*ln2hi ;
	118	z=hi-(lo=k*ln2lo);
	119	c=(hi-z)-lo;
	120
	121	if(k==0) return(z+exp__E(z,c));
	122	if(k==1)
	123	if(z< -0.25)
	124	{x=z+half;x +=exp__E(z,c); return(x+x);}
	125	else
	126	{z+=exp__E(z,c); x=half+z; return(x+x);}
	127	/* end of k=1 */
	128
	129	else {
	130	if(k<=prec)
	131	{ x=one-scalb(one,-k); z += exp__E(z,c);}
	132	else if(k<100)
	133	{ x = exp__E(z,c)-scalb(one,-k); x+=z; z=one;}
	134	else
	135	{ x = exp__E(z,c)+z; z=one;}
	136
	137	return (scalb(x+z,k));
	138	}
	139	}
	140	/* end of x > lnunfl */
	141
	142	else
	143	/* expm1(-big#) rounded to -1 (inexact) */
	144	if(finite(x))
	145	{ ln2hi+ln2lo; return(-one);}
	146
	147	/* expm1(-INF) is -1 */
	148	else return(-one);
	149	}
	150	/* end of x < lnhuge */
	151
	152	else
	153	/* expm1(INF) is INF, expm1(+big#) overflows to INF */
	154	return( finite(x) ? scalb(one,5000) : x);
	155	}