1 
2 /* @(#)e_exp.c 5.1 93/09/24 */
3 /*
4  * ====================================================
5  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6  *
7  * Developed at SunPro, a Sun Microsystems, Inc. business.
8  * Permission to use, copy, modify, and distribute this
9  * software is freely granted, provided that this notice
10  * is preserved.
11  * ====================================================
12  */
13 
14 /* __ieee754_exp(x)
15  * Returns the exponential of x.
16  *
17  * Method
18  *   1. Argument reduction:
19  *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
20  *	Given x, find r and integer k such that
21  *
22  *               x = k*ln2 + r,  |r| <= 0.5*ln2.
23  *
24  *      Here r will be represented as r = hi-lo for better
25  *	accuracy.
26  *
27  *   2. Approximation of exp(r) by a special rational function on
28  *	the interval [0,0.34658]:
29  *	Write
30  *	    R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
31  *      We use a special Reme algorithm on [0,0.34658] to generate
32  * 	a polynomial of degree 5 to approximate R. The maximum error
33  *	of this polynomial approximation is bounded by 2**-59. In
34  *	other words,
35  *	    R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
36  *  	(where z=r*r, and the values of P1 to P5 are listed below)
37  *	and
38  *	    |                  5          |     -59
39  *	    | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
40  *	    |                             |
41  *	The computation of exp(r) thus becomes
42  *                             2*r
43  *		exp(r) = 1 + -------
44  *		              R - r
45  *                                 r*R1(r)
46  *		       = 1 + r + ----------- (for better accuracy)
47  *		                  2 - R1(r)
48  *	where
49  *			         2       4             10
50  *		R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
51  *
52  *   3. Scale back to obtain exp(x):
53  *	From step 1, we have
54  *	   exp(x) = 2^k * exp(r)
55  *
56  * Special cases:
57  *	exp(INF) is INF, exp(NaN) is NaN;
58  *	exp(-INF) is 0, and
59  *	for finite argument, only exp(0)=1 is exact.
60  *
61  * Accuracy:
62  *	according to an error analysis, the error is always less than
63  *	1 ulp (unit in the last place).
64  *
65  * Misc. info.
66  *	For IEEE double
67  *	    if x >  7.09782712893383973096e+02 then exp(x) overflow
68  *	    if x < -7.45133219101941108420e+02 then exp(x) underflow
69  *
70  * Constants:
71  * The hexadecimal values are the intended ones for the following
72  * constants. The decimal values may be used, provided that the
73  * compiler will convert from decimal to binary accurately enough
74  * to produce the hexadecimal values shown.
75  */
76 
77 #include "fdlibm.h"
78 
79 #ifndef _DOUBLE_IS_32BITS
80 
81 #ifdef __STDC__
82 static const double
83 #else
84 static double
85 #endif
86 one	= 1.0,
87 halF[2]	= {0.5,-0.5,},
88 huge	= 1.0e+300,
89 twom1000= 9.33263618503218878990e-302,     /* 2**-1000=0x01700000,0*/
90 o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
91 u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */
92 ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */
93 	     -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
94 ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */
95 	     -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
96 invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
97 P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
98 P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
99 P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
100 P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
101 P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
102 
103 
104 #ifdef __STDC__
__ieee754_exp(double x)105 	double __ieee754_exp(double x)	/* default IEEE double exp */
106 #else
107 	double __ieee754_exp(x)	/* default IEEE double exp */
108 	double x;
109 #endif
110 {
111 	double y,hi,lo,c,t;
112 	int32_t k,xsb;
113 	uint32_t hx;
114 
115 	GET_HIGH_WORD(hx,x);
116 	xsb = (hx>>31)&1;		/* sign bit of x */
117 	hx &= 0x7fffffff;		/* high word of |x| */
118 
119     /* filter out non-finite argument */
120 	if(hx >= 0x40862E42) {			/* if |x|>=709.78... */
121             if(hx>=0x7ff00000) {
122 	        uint32_t lx;
123 		GET_LOW_WORD(lx,x);
124 		if(((hx&0xfffff)|lx)!=0)
125 		     return x+x; 		/* NaN */
126 		else return (xsb==0)? x:0.0;	/* exp(+-inf)={inf,0} */
127 	    }
128 	    if(x > o_threshold) return huge*huge; /* overflow */
129 	    if(x < u_threshold) return twom1000*twom1000; /* underflow */
130 	}
131 
132     /* argument reduction */
133 	if(hx > 0x3fd62e42) {		/* if  |x| > 0.5 ln2 */
134 	    if(hx < 0x3FF0A2B2) {	/* and |x| < 1.5 ln2 */
135 		hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
136 	    } else {
137 		k  = invln2*x+halF[xsb];
138 		t  = k;
139 		hi = x - t*ln2HI[0];	/* t*ln2HI is exact here */
140 		lo = t*ln2LO[0];
141 	    }
142 	    x  = hi - lo;
143 	}
144 	else if(hx < 0x3e300000)  {	/* when |x|<2**-28 */
145 	    if(huge+x>one) return one+x;/* trigger inexact */
146 	}
147 	else k = 0;
148 
149     /* x is now in primary range */
150 	t  = x*x;
151 	c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
152 	if(k==0) 	return one-((x*c)/(c-2.0)-x);
153 	else 		y = one-((lo-(x*c)/(2.0-c))-hi);
154 	if(k >= -1021) {
155 	    uint32_t hy;
156 	    GET_HIGH_WORD(hy,y);
157 	    SET_HIGH_WORD(y,hy+(k<<20));	/* add k to y's exponent */
158 	    return y;
159 	} else {
160 	    uint32_t hy;
161 	    GET_HIGH_WORD(hy,y);
162 	    SET_HIGH_WORD(y,hy+((k+1000)<<20));	/* add k to y's exponent */
163 	    return y*twom1000;
164 	}
165 }
166 
167 #endif /* defined(_DOUBLE_IS_32BITS) */
168