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/* __ieee754_exp(x)
14 * Returns the exponential of x.
15 *
16 * Method
17 *   1. Argument reduction:
18 *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
19 *	Given x, find r and integer k such that
20 *
21 *               x = k*ln2 + r,  |r| <= 0.5*ln2.
22 *
23 *      Here r will be represented as r = hi-lo for better
24 *	accuracy.
25 *
26 *   2. Approximation of ieee_exp(r) by a special rational function on
27 *	the interval [0,0.34658]:
28 *	Write
29 *	    R(r**2) = r*(ieee_exp(r)+1)/(ieee_exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
30 *      We use a special Remes algorithm on [0,0.34658] to generate
31 * 	a polynomial of degree 5 to approximate R. The maximum error
32 *	of this polynomial approximation is bounded by 2**-59. In
33 *	other words,
34 *	    R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
35 *  	(where z=r*r, and the values of P1 to P5 are listed below)
36 *	and
37 *	    |                  5          |     -59
38 *	    | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
39 *	    |                             |
40 *	The computation of ieee_exp(r) thus becomes
41 *                             2*r
42 *		exp(r) = 1 + -------
43 *		              R - r
44 *                                 r*R1(r)
45 *		       = 1 + r + ----------- (for better accuracy)
46 *		                  2 - R1(r)
47 *	where
48 *			         2       4             10
49 *		R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
50 *
51 *   3. Scale back to obtain ieee_exp(x):
52 *	From step 1, we have
53 *	   ieee_exp(x) = 2^k * ieee_exp(r)
54 *
55 * Special cases:
56 *	exp(INF) is INF, ieee_exp(NaN) is NaN;
57 *	exp(-INF) is 0, and
58 *	for finite argument, only ieee_exp(0)=1 is exact.
59 *
60 * Accuracy:
61 *	according to an error analysis, the error is always less than
62 *	1 ulp (unit in the last place).
63 *
64 * Misc. info.
65 *	For IEEE double
66 *	    if x >  7.09782712893383973096e+02 then ieee_exp(x) overflow
67 *	    if x < -7.45133219101941108420e+02 then ieee_exp(x) underflow
68 *
69 * Constants:
70 * The hexadecimal values are the intended ones for the following
71 * constants. The decimal values may be used, provided that the
72 * compiler will convert from decimal to binary accurately enough
73 * to produce the hexadecimal values shown.
74 */
75
76#include "fdlibm.h"
77
78#ifdef __STDC__
79static const double
80#else
81static double
82#endif
83one	= 1.0,
84halF[2]	= {0.5,-0.5,},
85huge	= 1.0e+300,
86twom1000= 9.33263618503218878990e-302,     /* 2**-1000=0x01700000,0*/
87o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
88u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */
89ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */
90	     -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
91ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */
92	     -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
93invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
94P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
95P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
96P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
97P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
98P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
99
100
101#ifdef __STDC__
102	double __ieee754_exp(double x)	/* default IEEE double exp */
103#else
104	double __ieee754_exp(x)	/* default IEEE double exp */
105	double x;
106#endif
107{
108	double y,hi,lo,c,t;
109	int k,xsb;
110	unsigned hx;
111
112	hx  = __HI(x);	/* high word of x */
113	xsb = (hx>>31)&1;		/* sign bit of x */
114	hx &= 0x7fffffff;		/* high word of |x| */
115
116    /* filter out non-finite argument */
117	if(hx >= 0x40862E42) {			/* if |x|>=709.78... */
118            if(hx>=0x7ff00000) {
119		if(((hx&0xfffff)|__LO(x))!=0)
120		     return x+x; 		/* NaN */
121		else return (xsb==0)? x:0.0;	/* ieee_exp(+-inf)={inf,0} */
122	    }
123	    if(x > o_threshold) return huge*huge; /* overflow */
124	    if(x < u_threshold) return twom1000*twom1000; /* underflow */
125	}
126
127    /* argument reduction */
128	if(hx > 0x3fd62e42) {		/* if  |x| > 0.5 ln2 */
129	    if(hx < 0x3FF0A2B2) {	/* and |x| < 1.5 ln2 */
130		hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
131	    } else {
132		k  = (int)(invln2*x+halF[xsb]);
133		t  = k;
134		hi = x - t*ln2HI[0];	/* t*ln2HI is exact here */
135		lo = t*ln2LO[0];
136	    }
137	    x  = hi - lo;
138	}
139	else if(hx < 0x3e300000)  {	/* when |x|<2**-28 */
140	    if(huge+x>one) return one+x;/* trigger inexact */
141	}
142	else k = 0;
143
144    /* x is now in primary range */
145	t  = x*x;
146	c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
147	if(k==0) 	return one-((x*c)/(c-2.0)-x);
148	else 		y = one-((lo-(x*c)/(2.0-c))-hi);
149	if(k >= -1021) {
150	    __HI(y) += (k<<20);	/* add k to y's exponent */
151	    return y;
152	} else {
153	    __HI(y) += ((k+1000)<<20);/* add k to y's exponent */
154	    return y*twom1000;
155	}
156}
157