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__
79 static const double
80 #else
81 static double
82 #endif
83 one	= 1.0,
84 halF[2]	= {0.5,-0.5,},
85 huge	= 1.0e+300,
86 twom1000= 9.33263618503218878990e-302,     /* 2**-1000=0x01700000,0*/
87 o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
88 u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */
89 ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */
90 	     -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
91 ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */
92 	     -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
93 invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
94 P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
95 P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
96 P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
97 P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
98 P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
99 
100 
101 #ifdef __STDC__
__ieee754_exp(double x)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