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("$FreeBSD$");
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 <float.h>
80 
81 #include "math.h"
82 #include "math_private.h"
83 
84 static const double
85 one	= 1.0,
86 halF[2]	= {0.5,-0.5,},
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 static volatile double
101 huge	= 1.0e+300,
102 twom1000= 9.33263618503218878990e-302;     /* 2**-1000=0x01700000,0*/
103 
104 double
__ieee754_exp(double x)105 __ieee754_exp(double x)	/* default IEEE double exp */
106 {
107 	double y,hi=0.0,lo=0.0,c,t,twopk;
108 	int32_t k=0,xsb;
109 	u_int32_t hx;
110 
111 	GET_HIGH_WORD(hx,x);
112 	xsb = (hx>>31)&1;		/* sign bit of x */
113 	hx &= 0x7fffffff;		/* high word of |x| */
114 
115     /* filter out non-finite argument */
116 	if(hx >= 0x40862E42) {			/* if |x|>=709.78... */
117             if(hx>=0x7ff00000) {
118 	        u_int32_t lx;
119 		GET_LOW_WORD(lx,x);
120 		if(((hx&0xfffff)|lx)!=0)
121 		     return x+x; 		/* NaN */
122 		else return (xsb==0)? x:0.0;	/* exp(+-inf)={inf,0} */
123 	    }
124 	    if(x > o_threshold) return huge*huge; /* overflow */
125 	    if(x < u_threshold) return twom1000*twom1000; /* underflow */
126 	}
127 
128     /* argument reduction */
129 	if(hx > 0x3fd62e42) {		/* if  |x| > 0.5 ln2 */
130 	    if(hx < 0x3FF0A2B2) {	/* and |x| < 1.5 ln2 */
131 		hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
132 	    } else {
133 		k  = (int)(invln2*x+halF[xsb]);
134 		t  = k;
135 		hi = x - t*ln2HI[0];	/* t*ln2HI is exact here */
136 		lo = t*ln2LO[0];
137 	    }
138 	    STRICT_ASSIGN(double, x, hi - lo);
139 	}
140 	else if(hx < 0x3e300000)  {	/* when |x|<2**-28 */
141 	    if(huge+x>one) return one+x;/* trigger inexact */
142 	}
143 	else k = 0;
144 
145     /* x is now in primary range */
146 	t  = x*x;
147 	if(k >= -1021)
148 	    INSERT_WORDS(twopk,0x3ff00000+(k<<20), 0);
149 	else
150 	    INSERT_WORDS(twopk,0x3ff00000+((k+1000)<<20), 0);
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 	    if (k==1024) return y*2.0*0x1p1023;
156 	    return y*twopk;
157 	} else {
158 	    return y*twopk*twom1000;
159 	}
160 }
161 
162 #ifdef SUPPORT_LONG_DOUBLE
163 #if (LDBL_MANT_DIG == 53)
164 __weak_reference(exp, expl);
165 #endif
166 #endif
167