e_exp.c revision 21673 
1/* @(#)e_exp.c 5.1 93/09/24 */
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
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#ifndef lint
14static char rcsid[] = "$FreeBSD: head/lib/msun/src/e_exp.c 21673 1997-01-14 07:20:47Z jkh $";
15#endif
16
17/* __ieee754_exp(x)
18 * Returns the exponential of x.
19 *
20 * Method
21 *   1. Argument reduction:
22 *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
23 *	Given x, find r and integer k such that
24 *
25 *               x = k*ln2 + r,  |r| <= 0.5*ln2.
26 *
27 *      Here r will be represented as r = hi-lo for better
28 *	accuracy.
29 *
30 *   2. Approximation of exp(r) by a special rational function on
31 *	the interval [0,0.34658]:
32 *	Write
33 *	    R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
34 *      We use a special Reme algorithm on [0,0.34658] to generate
35 * 	a polynomial of degree 5 to approximate R. The maximum error
36 *	of this polynomial approximation is bounded by 2**-59. In
37 *	other words,
38 *	    R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
39 *  	(where z=r*r, and the values of P1 to P5 are listed below)
40 *	and
41 *	    |                  5          |     -59
42 *	    | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
43 *	    |                             |
44 *	The computation of exp(r) thus becomes
45 *                             2*r
46 *		exp(r) = 1 + -------
47 *		              R - r
48 *                                 r*R1(r)
49 *		       = 1 + r + ----------- (for better accuracy)
50 *		                  2 - R1(r)
51 *	where
52 *			         2       4             10
53 *		R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
54 *
55 *   3. Scale back to obtain exp(x):
56 *	From step 1, we have
57 *	   exp(x) = 2^k * exp(r)
58 *
59 * Special cases:
60 *	exp(INF) is INF, exp(NaN) is NaN;
61 *	exp(-INF) is 0, and
62 *	for finite argument, only exp(0)=1 is exact.
63 *
64 * Accuracy:
65 *	according to an error analysis, the error is always less than
66 *	1 ulp (unit in the last place).
67 *
68 * Misc. info.
69 *	For IEEE double
70 *	    if x >  7.09782712893383973096e+02 then exp(x) overflow
71 *	    if x < -7.45133219101941108420e+02 then exp(x) underflow
72 *
73 * Constants:
74 * The hexadecimal values are the intended ones for the following
75 * constants. The decimal values may be used, provided that the
76 * compiler will convert from decimal to binary accurately enough
77 * to produce the hexadecimal values shown.
78 */
79
80#include "math.h"
81#include "math_private.h"
82
83#ifdef __STDC__
84static const double
85#else
86static double
87#endif
88one	= 1.0,
89halF[2]	= {0.5,-0.5,},
90huge	= 1.0e+300,
91twom1000= 9.33263618503218878990e-302,     /* 2**-1000=0x01700000,0*/
92o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
93u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */
94ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */
95	     -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
96ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */
97	     -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
98invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
99P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
100P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
101P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
102P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
103P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
104
105
106#ifdef __STDC__
107	double __ieee754_exp(double x)	/* default IEEE double exp */
108#else
109	double __ieee754_exp(x)	/* default IEEE double exp */
110	double x;
111#endif
112{
113	double y,hi=0.0,lo=0.0,c,t;
114	int32_t k=0,xsb;
115	u_int32_t hx;
116
117	GET_HIGH_WORD(hx,x);
118	xsb = (hx>>31)&1;		/* sign bit of x */
119	hx &= 0x7fffffff;		/* high word of |x| */
120
121    /* filter out non-finite argument */
122	if(hx >= 0x40862E42) {			/* if |x|>=709.78... */
123            if(hx>=0x7ff00000) {
124	        u_int32_t lx;
125		GET_LOW_WORD(lx,x);
126		if(((hx&0xfffff)|lx)!=0)
127		     return x+x; 		/* NaN */
128		else return (xsb==0)? x:0.0;	/* exp(+-inf)={inf,0} */
129	    }
130	    if(x > o_threshold) return huge*huge; /* overflow */
131	    if(x < u_threshold) return twom1000*twom1000; /* underflow */
132	}
133
134    /* argument reduction */
135	if(hx > 0x3fd62e42) {		/* if  |x| > 0.5 ln2 */
136	    if(hx < 0x3FF0A2B2) {	/* and |x| < 1.5 ln2 */
137		hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
138	    } else {
139		k  = invln2*x+halF[xsb];
140		t  = k;
141		hi = x - t*ln2HI[0];	/* t*ln2HI is exact here */
142		lo = t*ln2LO[0];
143	    }
144	    x  = hi - lo;
145	}
146	else if(hx < 0x3e300000)  {	/* when |x|<2**-28 */
147	    if(huge+x>one) return one+x;/* trigger inexact */
148	}
149	else k = 0;
150
151    /* x is now in primary range */
152	t  = x*x;
153	c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
154	if(k==0) 	return one-((x*c)/(c-2.0)-x);
155	else 		y = one-((lo-(x*c)/(2.0-c))-hi);
156	if(k >= -1021) {
157	    u_int32_t hy;
158	    GET_HIGH_WORD(hy,y);
159	    SET_HIGH_WORD(y,hy+(k<<20));	/* add k to y's exponent */
160	    return y;
161	} else {
162	    u_int32_t hy;
163	    GET_HIGH_WORD(hy,y);
164	    SET_HIGH_WORD(y,hy+((k+1000)<<20));	/* add k to y's exponent */
165	    return y*twom1000;
166	}
167}
168