Atlas - e_exp.c
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1/*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
10 */
11
12/* __ieee754_exp(x)
13 * Returns the exponential of x.
14 *
15 * Method
16 *   1. Argument reduction:
17 *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
18 *	Given x, find r and integer k such that
19 *
20 *               x = k*ln2 + r,  |r| <= 0.5*ln2.
21 *
22 *      Here r will be represented as r = hi-lo for better
23 *	accuracy.
24 *
25 *   2. Approximation of exp(r) by a special rational function on
26 *	the interval [0,0.34658]:
27 *	Write
28 *	    R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
29 *      We use a special Reme algorithm on [0,0.34658] to generate
30 * 	a polynomial of degree 5 to approximate R. The maximum error
31 *	of this polynomial approximation is bounded by 2**-59. In
32 *	other words,
33 *	    R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
34 *  	(where z=r*r, and the values of P1 to P5 are listed below)
35 *	and
36 *	    |                  5          |     -59
37 *	    | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
38 *	    |                             |
39 *	The computation of exp(r) thus becomes
40 *                             2*r
41 *		exp(r) = 1 + -------
42 *		              R - r
43 *                                 r*R1(r)
44 *		       = 1 + r + ----------- (for better accuracy)
45 *		                  2 - R1(r)
46 *	where
47 *			         2       4             10
48 *		R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
49 *
50 *   3. Scale back to obtain exp(x):
51 *	From step 1, we have
52 *	   exp(x) = 2^k * exp(r)
53 *
54 * Special cases:
55 *	exp(INF) is INF, exp(NaN) is NaN;
56 *	exp(-INF) is 0, and
57 *	for finite argument, only exp(0)=1 is exact.
58 *
59 * Accuracy:
60 *	according to an error analysis, the error is always less than
61 *	1 ulp (unit in the last place).
62 *
63 * Misc. info.
64 *	For IEEE double
65 *	    if x >  7.09782712893383973096e+02 then exp(x) overflow
66 *	    if x < -7.45133219101941108420e+02 then exp(x) underflow
67 *
68 * Constants:
69 * The hexadecimal values are the intended ones for the following
70 * constants. The decimal values may be used, provided that the
71 * compiler will convert from decimal to binary accurately enough
72 * to produce the hexadecimal values shown.
73 */
74
75#include "math_libm.h"
76#include "math_private.h"
77
78static const double
79one	= 1.0,
80halF[2]	= {0.5,-0.5,},
81huge	= 1.0e+300,
82twom1000= 9.33263618503218878990e-302,     /* 2**-1000=0x01700000,0*/
83o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
84u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */
85ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */
86	     -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
87ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */
88	     -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
89invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
90P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
91P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
92P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
93P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
94P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
95
96double __ieee754_exp(double x)	/* default IEEE double exp */
97{
98	double y;
99	double hi = 0.0;
100	double lo = 0.0;
101	double c;
102	double t;
103	int32_t k=0;
104	int32_t xsb;
105	u_int32_t hx;
106
107	GET_HIGH_WORD(hx,x);
108	xsb = (hx>>31)&1;		/* sign bit of x */
109	hx &= 0x7fffffff;		/* high word of |x| */
110
111    /* filter out non-finite argument */
112	if(hx >= 0x40862E42) {			/* if |x|>=709.78... */
113            if(hx>=0x7ff00000) {
114	        u_int32_t lx;
115		GET_LOW_WORD(lx,x);
116		if(((hx&0xfffff)|lx)!=0)
117		     return x+x; 		/* NaN */
118		else return (xsb==0)? x:0.0;	/* exp(+-inf)={inf,0} */
119	    }
120		#if 1
121		if(x > o_threshold) return huge*huge; /* overflow */
122		#else  /* !!! FIXME: check this: "huge * huge" is a compiler warning, maybe they wanted +Inf? */
123		if(x > o_threshold) return INFINITY; /* overflow */
124		#endif
125
126	    if(x < u_threshold) return twom1000*twom1000; /* underflow */
127	}
128
129    /* argument reduction */
130	if(hx > 0x3fd62e42) {		/* if  |x| > 0.5 ln2 */
131	    if(hx < 0x3FF0A2B2) {	/* and |x| < 1.5 ln2 */
132		hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
133	    } else {
134		k  = (int32_t) (invln2*x+halF[xsb]);
135		t  = k;
136		hi = x - t*ln2HI[0];	/* t*ln2HI is exact here */
137		lo = t*ln2LO[0];
138	    }
139	    x  = hi - lo;
140	}
141	else if(hx < 0x3e300000)  {	/* when |x|<2**-28 */
142	    if(huge+x>one) return one+x;/* trigger inexact */
143	}
144	else k = 0;
145
146    /* x is now in primary range */
147	t  = x*x;
148	c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
149	if(k==0) 	return one-((x*c)/(c-2.0)-x);
150	else 		y = one-((lo-(x*c)/(2.0-c))-hi);
151	if(k >= -1021) {
152	    u_int32_t hy;
153	    GET_HIGH_WORD(hy,y);
154	    SET_HIGH_WORD(y,hy+(k<<20));	/* add k to y's exponent */
155	    return y;
156	} else {
157	    u_int32_t hy;
158	    GET_HIGH_WORD(hy,y);
159	    SET_HIGH_WORD(y,hy+((k+1000)<<20));	/* add k to y's exponent */
160	    return y*twom1000;
161	}
162}
163
164/*
165 * wrapper exp(x)
166 */
167#ifndef _IEEE_LIBM
168double exp(double x)
169{
170	static const double o_threshold =  7.09782712893383973096e+02; /* 0x40862E42, 0xFEFA39EF */
171	static const double u_threshold = -7.45133219101941108420e+02; /* 0xc0874910, 0xD52D3051 */
172
173	double z = __ieee754_exp(x);
174	if (_LIB_VERSION == _IEEE_)
175		return z;
176	if (isfinite(x)) {
177		if (x > o_threshold)
178			return __kernel_standard(x, x, 6); /* exp overflow */
179		if (x < u_threshold)
180			return __kernel_standard(x, x, 7); /* exp underflow */
181	}
182	return z;
183}
184#else
185strong_alias(__ieee754_exp, exp)
186#endif
187libm_hidden_def(exp)
188
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