e_exp.c revision 176451 
1
2/* @(#)e_exp.c 1.6 04/04/22 */
3/*
4 * ====================================================
5 * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12
13#include <sys/cdefs.h>
14__FBSDID("$FreeBSD: head/lib/msun/src/e_exp.c 176451 2008-02-22 02:30:36Z das $");
15
16/* __ieee754_exp(x)
17 * Returns the exponential of x.
18 *
19 * Method
20 *   1. Argument reduction:
21 *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
22 *	Given x, find r and integer k such that
23 *
24 *               x = k*ln2 + r,  |r| <= 0.5*ln2.
25 *
26 *      Here r will be represented as r = hi-lo for better
27 *	accuracy.
28 *
29 *   2. Approximation of exp(r) by a special rational function on
30 *	the interval [0,0.34658]:
31 *	Write
32 *	    R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
33 *      We use a special Remes algorithm on [0,0.34658] to generate
34 * 	a polynomial of degree 5 to approximate R. The maximum error
35 *	of this polynomial approximation is bounded by 2**-59. In
36 *	other words,
37 *	    R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
38 *  	(where z=r*r, and the values of P1 to P5 are listed below)
39 *	and
40 *	    |                  5          |     -59
41 *	    | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
42 *	    |                             |
43 *	The computation of exp(r) thus becomes
44 *                             2*r
45 *		exp(r) = 1 + -------
46 *		              R - r
47 *                                 r*R1(r)
48 *		       = 1 + r + ----------- (for better accuracy)
49 *		                  2 - R1(r)
50 *	where
51 *			         2       4             10
52 *		R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
53 *
54 *   3. Scale back to obtain exp(x):
55 *	From step 1, we have
56 *	   exp(x) = 2^k * exp(r)
57 *
58 * Special cases:
59 *	exp(INF) is INF, exp(NaN) is NaN;
60 *	exp(-INF) is 0, and
61 *	for finite argument, only exp(0)=1 is exact.
62 *
63 * Accuracy:
64 *	according to an error analysis, the error is always less than
65 *	1 ulp (unit in the last place).
66 *
67 * Misc. info.
68 *	For IEEE double
69 *	    if x >  7.09782712893383973096e+02 then exp(x) overflow
70 *	    if x < -7.45133219101941108420e+02 then exp(x) underflow
71 *
72 * Constants:
73 * The hexadecimal values are the intended ones for the following
74 * constants. The decimal values may be used, provided that the
75 * compiler will convert from decimal to binary accurately enough
76 * to produce the hexadecimal values shown.
77 */
78
79#include "math.h"
80#include "math_private.h"
81
82static const double
83one	= 1.0,
84halF[2]	= {0.5,-0.5,},
85huge	= 1.0e+300,
86o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
87u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */
88ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */
89	     -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
90ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */
91	     -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
92invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
93P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
94P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
95P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
96P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
97P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
98
99static volatile double
100twom1000= 9.33263618503218878990e-302;     /* 2**-1000=0x01700000,0*/
101
102double
103__ieee754_exp(double x)	/* default IEEE double exp */
104{
105	double y,hi=0.0,lo=0.0,c,t,twopk;
106	int32_t k=0,xsb;
107	u_int32_t hx;
108
109	GET_HIGH_WORD(hx,x);
110	xsb = (hx>>31)&1;		/* sign bit of x */
111	hx &= 0x7fffffff;		/* high word of |x| */
112
113    /* filter out non-finite argument */
114	if(hx >= 0x40862E42) {			/* if |x|>=709.78... */
115            if(hx>=0x7ff00000) {
116	        u_int32_t lx;
117		GET_LOW_WORD(lx,x);
118		if(((hx&0xfffff)|lx)!=0)
119		     return x+x; 		/* NaN */
120		else return (xsb==0)? x:0.0;	/* exp(+-inf)={inf,0} */
121	    }
122	    if(x > o_threshold) return huge*huge; /* overflow */
123	    if(x < u_threshold) return twom1000*twom1000; /* underflow */
124	}
125
126    /* argument reduction */
127	if(hx > 0x3fd62e42) {		/* if  |x| > 0.5 ln2 */
128	    if(hx < 0x3FF0A2B2) {	/* and |x| < 1.5 ln2 */
129		hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
130	    } else {
131		k  = (int)(invln2*x+halF[xsb]);
132		t  = k;
133		hi = x - t*ln2HI[0];	/* t*ln2HI is exact here */
134		lo = t*ln2LO[0];
135	    }
136	    x  = hi - lo;
137	}
138	else if(hx < 0x3e300000)  {	/* when |x|<2**-28 */
139	    if(huge+x>one) return one+x;/* trigger inexact */
140	}
141	else k = 0;
142
143    /* x is now in primary range */
144	t  = x*x;
145	if(k >= -1021)
146	    INSERT_WORDS(twopk,0x3ff00000+(k<<20), 0);
147	else
148	    INSERT_WORDS(twopk,0x3ff00000+((k+1000)<<20), 0);
149	c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
150	if(k==0) 	return one-((x*c)/(c-2.0)-x);
151	else 		y = one-((lo-(x*c)/(2.0-c))-hi);
152	if(k >= -1021) {
153	    if (k==1024) return y*2.0*0x1p1023;
154	    return y*twopk;
155	} else {
156	    return y*twopk*twom1000;
157	}
158}
159