lib/libm/src/b_exp.c

/*
 * Copyright (c) 1985, 1993
 *	The Regents of the University of California.  All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 * 3. All advertising materials mentioning features or use of this software
 *    must display the following acknowledgement:
 *	This product includes software developed by the University of
 *	California, Berkeley and its contributors.
 * 4. Neither the name of the University nor the names of its contributors
 *    may be used to endorse or promote products derived from this software
 *    without specific prior written permission.
 *
 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
 * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
 * SUCH DAMAGE.
 *
 * FreeBSD SVN: 226414 (2011-10-16)
 */

/* @(#)exp.c	8.1 (Berkeley) 6/4/93 */

/* EXP(X)
 * RETURN THE EXPONENTIAL OF X
 * 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, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
 *
 * Required system supported functions:
 *	scalb(x,n)
 *	copysign(x,y)
 *	finite(x)
 *
 * 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 exp(r) by
 *
 *		exp(r) = 1 + r + r*R1/(2-R1),
 *	   where
 *		R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
 *
 *	3. exp(x) = 2^k * exp(r) .
 *
 * Special cases:
 *	exp(INF) is INF, exp(NaN) is NaN;
 *	exp(-INF)=  0;
 *	for finite argument, only exp(0)=1 is exact.
 *
 * Accuracy:
 *	exp(x) returns the exponential of x nearly rounded. In a test run
 *	with 1,156,000 random arguments on a VAX, the maximum observed
 *	error was 0.869 ulps (units in the last place).
 */

#include "mathimpl.h"

static const double p1 = 0x1.555555555553ep-3;
static const double p2 = -0x1.6c16c16bebd93p-9;
static const double p3 = 0x1.1566aaf25de2cp-14;
static const double p4 = -0x1.bbd41c5d26bf1p-20;
static const double p5 = 0x1.6376972bea4d0p-25;
static const double ln2hi = 0x1.62e42fee00000p-1;
static const double ln2lo = 0x1.a39ef35793c76p-33;
static const double lnhuge = 0x1.6602b15b7ecf2p9;
static const double lntiny = -0x1.77af8ebeae354p9;
static const double invln2 = 0x1.71547652b82fep0;


/* returns exp(r = x + c) for |c| < |x| with no overlap.  */

double
__exp__D(double x, double c)
{
	double  z,hi,lo;
	int k;

	if (x != x)	/* x is NaN */
		return(x);
	if ( x <= lnhuge ) {
		if ( x >= lntiny ) {

		    /* argument reduction : x --> x - k*ln2 */
			z = invln2*x;
			k = z + copysign(.5, x);

		    /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */

			hi=(x-k*ln2hi);			/* Exact. */
			x= hi - (lo = k*ln2lo-c);
		    /* return 2^k*[1+x+x*c/(2+c)]  */
			z=x*x;
			c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
			c = (x*c)/(2.0-c);

			return  scalb(1.+(hi-(lo - c)), k);
		}
		/* end of x > lntiny */

		else
		     /* exp(-big#) underflows to zero */
		     if(finite(x))  return(scalb(1.0,-5000));

		     /* exp(-INF) is zero */
		     else return(0.0);
	}
	/* end of x < lnhuge */

	else
	/* exp(INF) is INF, exp(+big#) overflows to INF */
	    return( finite(x) ?  scalb(1.0,5000)  : x);
}
Commit	Line	Data
	1	/*
	2	* Copyright (c) 1985, 1993
	3	* The Regents of the University of California. All rights reserved.
	4	*
	5	* Redistribution and use in source and binary forms, with or without
	6	* modification, are permitted provided that the following conditions
	7	* are met:
	8	* 1. Redistributions of source code must retain the above copyright
	9	* notice, this list of conditions and the following disclaimer.
	10	* 2. Redistributions in binary form must reproduce the above copyright
	11	* notice, this list of conditions and the following disclaimer in the
	12	* documentation and/or other materials provided with the distribution.
	13	* 3. All advertising materials mentioning features or use of this software
	14	* must display the following acknowledgement:
	15	* This product includes software developed by the University of
	16	* California, Berkeley and its contributors.
	17	* 4. Neither the name of the University nor the names of its contributors
	18	* may be used to endorse or promote products derived from this software
	19	* without specific prior written permission.
	20	*
	21	* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
	22	* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
	23	* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
	24	* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
	25	* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
	26	* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
	27	* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
	28	* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
	29	* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
	30	* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
	31	* SUCH DAMAGE.
	32	*
	33	* FreeBSD SVN: 226414 (2011-10-16)
	34	*/
	35
	36	/* @(#)exp.c 8.1 (Berkeley) 6/4/93 */
	37
	38	/* EXP(X)
	39	* RETURN THE EXPONENTIAL OF X
	40	* DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
	41	* CODED IN C BY K.C. NG, 1/19/85;
	42	* REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
	43	*
	44	* Required system supported functions:
	45	* scalb(x,n)
	46	* copysign(x,y)
	47	* finite(x)
	48	*
	49	* Method:
	50	* 1. Argument Reduction: given the input x, find r and integer k such
	51	* that
	52	* x = kln2 + r, \|r\| <= 0.5ln2 .
	53	* r will be represented as r := z+c for better accuracy.
	54	*
	55	* 2. Compute exp(r) by
	56	*
	57	* exp(r) = 1 + r + r*R1/(2-R1),
	58	* where
	59	* R1 = x - x^2(p1+x^2(p2+x^2(p3+x^2(p4+p5*x^2)))).
	60	*
	61	* 3. exp(x) = 2^k * exp(r) .
	62	*
	63	* Special cases:
	64	* exp(INF) is INF, exp(NaN) is NaN;
	65	* exp(-INF)= 0;
	66	* for finite argument, only exp(0)=1 is exact.
	67	*
	68	* Accuracy:
	69	* exp(x) returns the exponential of x nearly rounded. In a test run
	70	* with 1,156,000 random arguments on a VAX, the maximum observed
	71	* error was 0.869 ulps (units in the last place).
	72	*/
	73
	74	#include "mathimpl.h"
	75
	76	static const double p1 = 0x1.555555555553ep-3;
	77	static const double p2 = -0x1.6c16c16bebd93p-9;
	78	static const double p3 = 0x1.1566aaf25de2cp-14;
	79	static const double p4 = -0x1.bbd41c5d26bf1p-20;
	80	static const double p5 = 0x1.6376972bea4d0p-25;
	81	static const double ln2hi = 0x1.62e42fee00000p-1;
	82	static const double ln2lo = 0x1.a39ef35793c76p-33;
	83	static const double lnhuge = 0x1.6602b15b7ecf2p9;
	84	static const double lntiny = -0x1.77af8ebeae354p9;
	85	static const double invln2 = 0x1.71547652b82fep0;
	86
	87
	88	/* returns exp(r = x + c) for \|c\| < \|x\| with no overlap. */
	89
	90	double
	91	__exp__D(double x, double c)
	92	{
	93	double z,hi,lo;
	94	int k;
	95
	96	if (x != x) /* x is NaN */
	97	return(x);
	98	if ( x <= lnhuge ) {
	99	if ( x >= lntiny ) {
	100
	101	/* argument reduction : x --> x - kln2 /
	102	z = invln2*x;
	103	k = z + copysign(.5, x);
	104
	105	/* express (x+c)-kln2 as hi-lo and let x=hi-lo rounded /
	106
	107	hi=(x-kln2hi); / Exact. */
	108	x= hi - (lo = k*ln2lo-c);
	109	/* return 2^k[1+x+xc/(2+c)] */
	110	z=x*x;
	111	c= x - z(p1+z(p2+z(p3+z(p4+z*p5))));
	112	c = (x*c)/(2.0-c);
	113
	114	return scalb(1.+(hi-(lo - c)), k);
	115	}
	116	/* end of x > lntiny */
	117
	118	else
	119	/* exp(-big#) underflows to zero */
	120	if(finite(x)) return(scalb(1.0,-5000));
	121
	122	/* exp(-INF) is zero */
	123	else return(0.0);
	124	}
	125	/* end of x < lnhuge */
	126
	127	else
	128	/* exp(INF) is INF, exp(+big#) overflows to INF */
	129	return( finite(x) ? scalb(1.0,5000) : x);
	130	}