[unix-history] / usr / src / lib / libm / common_source / pow.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[] =
"@(#)pow.c	4.5 (Berkeley) 8/21/85; 1.7 (ucb.elefunt) %G%";
#endif	/* not lint */

/* POW(X,Y)  
 * RETURN X**Y 
 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
 * CODED IN C BY K.C. NG, 1/8/85; 
 * REVISED BY K.C. NG on 7/10/85.
 *
 * Required system supported functions:
 *      scalb(x,n)      
 *      logb(x)         
 *	copysign(x,y)	
 *	finite(x)	
 *	drem(x,y)
 *
 * Required kernel functions:
 *	exp__E(a,c)	...return  exp(a+c) - 1 - a*a/2
 *	log__L(x)	...return  (log(1+x) - 2s)/s, s=x/(2+x) 
 *	pow_p(x,y)	...return  +(anything)**(finite non zero)
 *
 * Method
 *	1. Compute and return log(x) in three pieces:
 *		log(x) = n*ln2 + hi + lo,
 *	   where n is an integer.
 *	2. Perform y*log(x) by simulating muti-precision arithmetic and 
 *	   return the answer in three pieces:
 *		y*log(x) = m*ln2 + hi + lo,
 *	   where m is an integer.
 *	3. Return x**y = exp(y*log(x))
 *		= 2^m * ( exp(hi+lo) ).
 *
 * Special cases:
 *	(anything) ** 0  is 1 ;
 *	(anything) ** 1  is itself;
 *	(anything) ** NaN is NaN;
 *	NaN ** (anything except 0) is NaN;
 *	+-(anything > 1) ** +INF is +INF;
 *	+-(anything > 1) ** -INF is +0;
 *	+-(anything < 1) ** +INF is +0;
 *	+-(anything < 1) ** -INF is +INF;
 *	+-1 ** +-INF is NaN and signal INVALID;
 *	+0 ** +(anything except 0, NaN)  is +0;
 *	-0 ** +(anything except 0, NaN, odd integer)  is +0;
 *	+0 ** -(anything except 0, NaN)  is +INF and signal DIV-BY-ZERO;
 *	-0 ** -(anything except 0, NaN, odd integer)  is +INF with signal;
 *	-0 ** (odd integer) = -( +0 ** (odd integer) );
 *	+INF ** +(anything except 0,NaN) is +INF;
 *	+INF ** -(anything except 0,NaN) is +0;
 *	-INF ** (odd integer) = -( +INF ** (odd integer) );
 *	-INF ** (even integer) = ( +INF ** (even integer) );
 *	-INF ** -(anything except integer,NaN) is NaN with signal;
 *	-(x=anything) ** (k=integer) is (-1)**k * (x ** k);
 *	-(anything except 0) ** (non-integer) is NaN with signal;
 *
 * Accuracy:
 *	pow(x,y) returns x**y nearly rounded. In particular, on a SUN, a VAX,
 *	and a Zilog Z8000,
 *			pow(integer,integer)
 *	always returns the correct integer provided it is representable.
 *	In a test run with 100,000 random arguments with 0 < x, y < 20.0
 *	on a VAX, the maximum observed error was 1.79 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.
 */

#if defined(vax)||defined(tahoe)	/* VAX D format */
#include <errno.h>
extern double infnan();
#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 */
/* invln2 =  1.4426950408889634148E0     , Hex  2^  1   *  .B8AA3B295C17F1 */
/* sqrt2  =  1.4142135623730950622E0     ; Hex  2^  1   *  .B504F333F9DE65 */
static long     ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)};
static long     ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)};
static long    invln2x[] = { _0x(aa3b,40b8), _0x(17f1,295c)};
static long     sqrt2x[] = { _0x(04f3,40b5), _0x(de65,33f9)};
#define    ln2hi    (*(double*)ln2hix)
#define    ln2lo    (*(double*)ln2lox)
#define   invln2    (*(double*)invln2x)
#define    sqrt2    (*(double*)sqrt2x)
#else	/* defined(vax)||defined(tahoe)	*/
static double
ln2hi  =  6.9314718036912381649E-1    , /*Hex  2^ -1   *  1.62E42FEE00000 */
ln2lo  =  1.9082149292705877000E-10   , /*Hex  2^-33   *  1.A39EF35793C76 */
invln2 =  1.4426950408889633870E0     , /*Hex  2^  0   *  1.71547652B82FE */
sqrt2  =  1.4142135623730951455E0     ; /*Hex  2^  0   *  1.6A09E667F3BCD */
#endif	/* defined(vax)||defined(tahoe)	*/

static double zero=0.0, half=1.0/2.0, one=1.0, two=2.0, negone= -1.0;

double pow(x,y)  	
double x,y;
{
	double drem(),pow_p(),copysign(),t;
	int finite();

	if     (y==zero)      return(one);
	else if(y==one
#if !defined(vax)&&!defined(tahoe)
		||x!=x
#endif	/* !defined(vax)&&!defined(tahoe) */
		) return( x );      /* if x is NaN or y=1 */
#if !defined(vax)&&!defined(tahoe)
	else if(y!=y)         return( y );      /* if y is NaN */
#endif	/* !defined(vax)&&!defined(tahoe) */
	else if(!finite(y))                     /* if y is INF */
	     if((t=copysign(x,one))==one) return(zero/zero);
	     else if(t>one) return((y>zero)?y:zero);
	     else return((y<zero)?-y:zero);
	else if(y==two)       return(x*x);
	else if(y==negone)    return(one/x);

    /* sign(x) = 1 */
	else if(copysign(one,x)==one) return(pow_p(x,y));

    /* sign(x)= -1 */
	/* if y is an even integer */
	else if ( (t=drem(y,two)) == zero)	return( pow_p(-x,y) );

	/* if y is an odd integer */
	else if (copysign(t,one) == one) return( -pow_p(-x,y) );

	/* Henceforth y is not an integer */
	else if(x==zero)	/* x is -0 */
	    return((y>zero)?-x:one/(-x));
	else {			/* return NaN */
#if defined(vax)||defined(tahoe)
	    return (infnan(EDOM));	/* NaN */
#else	/* defined(vax)||defined(tahoe) */
	    return(zero/zero);
#endif	/* defined(vax)||defined(tahoe) */
	}
}

/* pow_p(x,y) return x**y for x with sign=1 and finite y */
static double pow_p(x,y)       
double x,y;
{
        double logb(),scalb(),copysign(),log__L(),exp__E();
        double c,s,t,z,tx,ty;
#ifdef tahoe
	double tahoe_tmp;
#endif	/* tahoe */
        float sx,sy;
	long k=0;
        int n,m;

	if(x==zero||!finite(x)) {           /* if x is +INF or +0 */
#if defined(vax)||defined(tahoe)
	     return((y>zero)?x:infnan(ERANGE));	/* if y<zero, return +INF */
#else	/* defined(vax)||defined(tahoe) */
	     return((y>zero)?x:one/x);
#endif	/* defined(vax)||defined(tahoe) */
	}
	if(x==1.0) return(x);	/* if x=1.0, return 1 since y is finite */

    /* reduce x to z in [sqrt(1/2)-1, sqrt(2)-1] */
        z=scalb(x,-(n=logb(x)));  
#if !defined(vax)&&!defined(tahoe)	/* IEEE double; subnormal number */
        if(n <= -1022) {n += (m=logb(z)); z=scalb(z,-m);} 
#endif	/* !defined(vax)&&!defined(tahoe) */
        if(z >= sqrt2 ) {n += 1; z *= half;}  z -= one ;

    /* log(x) = nlog2+log(1+z) ~ nlog2 + t + tx */
	s=z/(two+z); c=z*z*half; tx=s*(c+log__L(s*s)); 
	t= z-(c-tx); tx += (z-t)-c;

   /* if y*log(x) is neither too big nor too small */
	if((s=logb(y)+logb(n+t)) < 12.0) 
	    if(s>-60.0) {

	/* compute y*log(x) ~ mlog2 + t + c */
        	s=y*(n+invln2*t);
                m=s+copysign(half,s);   /* m := nint(y*log(x)) */ 
		k=y; 
		if((double)k==y) {	/* if y is an integer */
		    k = m-k*n;
		    sx=t; tx+=(t-sx); }
		else	{		/* if y is not an integer */    
		    k =m;
	 	    tx+=n*ln2lo;
		    sx=(c=n*ln2hi)+t; tx+=(c-sx)+t; }
	   /* end of checking whether k==y */

                sy=y; ty=y-sy;          /* y ~ sy + ty */
#ifdef tahoe
		s = (tahoe_tmp = sx)*sy-k*ln2hi;
#else	/* tahoe */
		s=(double)sx*sy-k*ln2hi;        /* (sy+ty)*(sx+tx)-kln2 */
#endif	/* tahoe */
		z=(tx*ty-k*ln2lo);
		tx=tx*sy; ty=sx*ty;
		t=ty+z; t+=tx; t+=s;
		c= -((((t-s)-tx)-ty)-z);

	    /* return exp(y*log(x)) */
		t += exp__E(t,c); return(scalb(one+t,m));
	     }
	/* end of if log(y*log(x)) > -60.0 */
	    
	    else
		/* exp(+- tiny) = 1 with inexact flag */
			{ln2hi+ln2lo; return(one);}
	    else if(copysign(one,y)*(n+invln2*t) <zero)
		/* exp(-(big#)) underflows to zero */
	        	return(scalb(one,-5000)); 
	    else
	        /* exp(+(big#)) overflows to INF */
	    		return(scalb(one, 5000)); 

}
Commit	Line	Data
9f4a7cc1 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[] =
859dc438 ZAL	16	"@(#)pow.c 4.5 (Berkeley) 8/21/85; 1.7 (ucb.elefunt) %G%";
859dc438 ZAL	17	#endif /* not lint */
9f4a7cc1 ZAL	18
	19	/* POW(X,Y)
	20	* RETURN X**Y
	21	* DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
	22	* CODED IN C BY K.C. NG, 1/8/85;
	23	* REVISED BY K.C. NG on 7/10/85.
	24	*
	25	* Required system supported functions:
	26	* scalb(x,n)
	27	* logb(x)
	28	* copysign(x,y)
	29	* finite(x)
	30	* drem(x,y)
	31	*
	32	* Required kernel functions:
	33	* exp__E(a,c) ...return exp(a+c) - 1 - a*a/2
	34	* log__L(x) ...return (log(1+x) - 2s)/s, s=x/(2+x)
	35	* pow_p(x,y) ...return +(anything)**(finite non zero)
	36	*
	37	* Method
	38	* 1. Compute and return log(x) in three pieces:
	39	* log(x) = n*ln2 + hi + lo,
	40	* where n is an integer.
	41	* 2. Perform y*log(x) by simulating muti-precision arithmetic and
	42	* return the answer in three pieces:
	43	* ylog(x) = mln2 + hi + lo,
	44	* where m is an integer.
	45	* 3. Return x*y = exp(ylog(x))
	46	* = 2^m * ( exp(hi+lo) ).
	47	*
	48	* Special cases:
	49	* (anything) ** 0 is 1 ;
	50	* (anything) ** 1 is itself;
	51	* (anything) ** NaN is NaN;
	52	* NaN ** (anything except 0) is NaN;
	53	* +-(anything > 1) ** +INF is +INF;
	54	* +-(anything > 1) ** -INF is +0;
	55	* +-(anything < 1) ** +INF is +0;
	56	* +-(anything < 1) ** -INF is +INF;
	57	* +-1 ** +-INF is NaN and signal INVALID;
	58	* +0 ** +(anything except 0, NaN) is +0;
	59	* -0 ** +(anything except 0, NaN, odd integer) is +0;
	60	* +0 ** -(anything except 0, NaN) is +INF and signal DIV-BY-ZERO;
	61	* -0 ** -(anything except 0, NaN, odd integer) is +INF with signal;
	62	* -0 (odd integer) = -( +0 (odd integer) );
	63	* +INF ** +(anything except 0,NaN) is +INF;
	64	* +INF ** -(anything except 0,NaN) is +0;
	65	* -INF (odd integer) = -( +INF (odd integer) );
	66	* -INF (even integer) = ( +INF (even integer) );
	67	* -INF ** -(anything except integer,NaN) is NaN with signal;
	68	* -(x=anything) (k=integer) is (-1)k * (x ** k);
	69	* -(anything except 0) ** (non-integer) is NaN with signal;
	70	*
	71	* Accuracy:
	72	* pow(x,y) returns x**y nearly rounded. In particular, on a SUN, a VAX,
	73	* and a Zilog Z8000,
	74	* pow(integer,integer)
	75	* always returns the correct integer provided it is representable.
	76	* In a test run with 100,000 random arguments with 0 < x, y < 20.0
	77	* on a VAX, the maximum observed error was 1.79 ulps (units in the
	78	* last place).
	79	*
	80	* Constants :
	81	* The hexadecimal values are the intended ones for the following constants.
82	* The decimal values may be used, provided that the compiler will convert
83	* from decimal to binary accurately enough to produce the hexadecimal values
84	* shown.
85	*/
86
859dc438	87	#if defined(vax)\|\|defined(tahoe) /* VAX D format */
9f4a7cc1 ZAL	88	#include <errno.h>
9f4a7cc1 ZAL	89	extern double infnan();
859dc438	90	#ifdef vax
0e01cbea	91	#define _0x(A,B) 0x//A//B
859dc438	92	#else /* vax */
0e01cbea	93	#define _0x(A,B) 0x//B//A
859dc438	94	#endif /* vax */
62b65e15	95	/* static double */
9f4a7cc1 ZAL	96	/* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */
	97	/* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */
	98	/* invln2 = 1.4426950408889634148E0 , Hex 2^ 1 * .B8AA3B295C17F1 */
	99	/* sqrt2 = 1.4142135623730950622E0 ; Hex 2^ 1 * .B504F333F9DE65 */
0e01cbea ZAL	100	static long ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)};
	101	static long ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)};
	102	static long invln2x[] = { _0x(aa3b,40b8), _0x(17f1,295c)};
	103	static long sqrt2x[] = { _0x(04f3,40b5), _0x(de65,33f9)};
9f4a7cc1 ZAL	104	#define ln2hi ((double)ln2hix)
	105	#define ln2lo ((double)ln2lox)
	106	#define invln2 ((double)invln2x)
	107	#define sqrt2 ((double)sqrt2x)
859dc438	108	#else /* defined(vax)\|\|defined(tahoe) */
62b65e15	109	static double
9f4a7cc1 ZAL	110	ln2hi = 6.9314718036912381649E-1 , /Hex 2^ -1 1.62E42FEE00000 */
	111	ln2lo = 1.9082149292705877000E-10 , /Hex 2^-33 1.A39EF35793C76 */
	112	invln2 = 1.4426950408889633870E0 , /Hex 2^ 0 1.71547652B82FE */
	113	sqrt2 = 1.4142135623730951455E0 ; /Hex 2^ 0 1.6A09E667F3BCD */
859dc438	114	#endif /* defined(vax)\|\|defined(tahoe) */
9f4a7cc1	115
62b65e15	116	static double zero=0.0, half=1.0/2.0, one=1.0, two=2.0, negone= -1.0;
9f4a7cc1 ZAL	117
	118	double pow(x,y)
	119	double x,y;
	120	{
	121	double drem(),pow_p(),copysign(),t;
	122	int finite();
	123
	124	if (y==zero) return(one);
	125	else if(y==one
859dc438	126	#if !defined(vax)&&!defined(tahoe)
9f4a7cc1	127	\|\|x!=x
859dc438	128	#endif /* !defined(vax)&&!defined(tahoe) */
9f4a7cc1	129	) return( x ); /* if x is NaN or y=1 */
859dc438	130	#if !defined(vax)&&!defined(tahoe)
9f4a7cc1	131	else if(y!=y) return( y ); /* if y is NaN */
859dc438	132	#endif /* !defined(vax)&&!defined(tahoe) */
9f4a7cc1 ZAL	133	else if(!finite(y)) /* if y is INF */
	134	if((t=copysign(x,one))==one) return(zero/zero);
	135	else if(t>one) return((y>zero)?y:zero);
	136	else return((y<zero)?-y:zero);
	137	else if(y==two) return(x*x);
	138	else if(y==negone) return(one/x);
	139
	140	/* sign(x) = 1 */
	141	else if(copysign(one,x)==one) return(pow_p(x,y));
	142
	143	/* sign(x)= -1 */
	144	/* if y is an even integer */
	145	else if ( (t=drem(y,two)) == zero) return( pow_p(-x,y) );
	146
	147	/* if y is an odd integer */
	148	else if (copysign(t,one) == one) return( -pow_p(-x,y) );
	149
	150	/* Henceforth y is not an integer */
	151	else if(x==zero) /* x is -0 */
	152	return((y>zero)?-x:one/(-x));
	153	else { /* return NaN */
859dc438	154	#if defined(vax)\|\|defined(tahoe)
9f4a7cc1	155	return (infnan(EDOM)); /* NaN */
859dc438	156	#else /* defined(vax)\|\|defined(tahoe) */
9f4a7cc1	157	return(zero/zero);
859dc438	158	#endif /* defined(vax)\|\|defined(tahoe) */
9f4a7cc1 ZAL	159	}
	160	}
	161
	162	/* pow_p(x,y) return x*y for x with sign=1 and finite y /
	163	static double pow_p(x,y)
	164	double x,y;
	165	{
	166	double logb(),scalb(),copysign(),log__L(),exp__E();
	167	double c,s,t,z,tx,ty;
859dc438	168	#ifdef tahoe
471c7555	169	double tahoe_tmp;
859dc438	170	#endif /* tahoe */
9f4a7cc1 ZAL	171	float sx,sy;
	172	long k=0;
	173	int n,m;
	174
	175	if(x==zero\|\|!finite(x)) { /* if x is +INF or +0 */
859dc438	176	#if defined(vax)\|\|defined(tahoe)
9f4a7cc1	177	return((y>zero)?x:infnan(ERANGE)); /* if y<zero, return +INF */
859dc438	178	#else /* defined(vax)\|\|defined(tahoe) */
9f4a7cc1	179	return((y>zero)?x:one/x);
859dc438	180	#endif /* defined(vax)\|\|defined(tahoe) */
9f4a7cc1 ZAL	181	}
	182	if(x==1.0) return(x); /* if x=1.0, return 1 since y is finite */
	183
	184	/* reduce x to z in [sqrt(1/2)-1, sqrt(2)-1] */
	185	z=scalb(x,-(n=logb(x)));
859dc438	186	#if !defined(vax)&&!defined(tahoe) /* IEEE double; subnormal number */
9f4a7cc1	187	if(n <= -1022) {n += (m=logb(z)); z=scalb(z,-m);}
859dc438	188	#endif /* !defined(vax)&&!defined(tahoe) */
9f4a7cc1 ZAL	189	if(z >= sqrt2 ) {n += 1; z *= half;} z -= one ;
	190
	191	/* log(x) = nlog2+log(1+z) ~ nlog2 + t + tx */
	192	s=z/(two+z); c=zzhalf; tx=s(c+log__L(ss));
	193	t= z-(c-tx); tx += (z-t)-c;
	194
	195	/* if ylog(x) is neither too big nor too small /
	196	if((s=logb(y)+logb(n+t)) < 12.0)
	197	if(s>-60.0) {
	198
	199	/* compute ylog(x) ~ mlog2 + t + c /
	200	s=y(n+invln2t);
	201	m=s+copysign(half,s); /* m := nint(ylog(x)) /
	202	k=y;
	203	if((double)k==y) { /* if y is an integer */
	204	k = m-k*n;
	205	sx=t; tx+=(t-sx); }
	206	else { /* if y is not an integer */
	207	k =m;
	208	tx+=n*ln2lo;
	209	sx=(c=n*ln2hi)+t; tx+=(c-sx)+t; }
	210	/* end of checking whether k==y */
	211
	212	sy=y; ty=y-sy; /* y ~ sy + ty */
859dc438	213	#ifdef tahoe
471c7555	214	s = (tahoe_tmp = sx)sy-kln2hi;
859dc438	215	#else /* tahoe */
9f4a7cc1	216	s=(double)sxsy-kln2hi; /* (sy+ty)(sx+tx)-kln2 /
859dc438	217	#endif /* tahoe */
9f4a7cc1 ZAL	218	z=(txty-kln2lo);
	219	tx=txsy; ty=sxty;
	220	t=ty+z; t+=tx; t+=s;
	221	c= -((((t-s)-tx)-ty)-z);
	222
	223	/* return exp(ylog(x)) /
	224	t += exp__E(t,c); return(scalb(one+t,m));
	225	}
	226	/* end of if log(ylog(x)) > -60.0 /
	227
	228	else
	229	/* exp(+- tiny) = 1 with inexact flag */
	230	{ln2hi+ln2lo; return(one);}
	231	else if(copysign(one,y)(n+invln2t) <zero)
	232	/* exp(-(big#)) underflows to zero */
	233	return(scalb(one,-5000));
	234	else
	235	/* exp(+(big#)) overflows to INF */
	236	return(scalb(one, 5000));
	237
	238	}