source: trunk/sys/libm/e_exp.c @ 1

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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 <libm/fdlibm.h>
78
79#ifdef __STDC__
80static const double
81#else
82static double
83#endif
84one     = 1.0,
85halF[2] = {0.5,-0.5,},
86huge    = 1.0e+300,
87twom1000= 9.33263618503218878990e-302,     /* 2**-1000=0x01700000,0*/
88o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
89u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */
90ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */
91             -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
92ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */
93             -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
94invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
95P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
96P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
97P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
98P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
99P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
100
101
102#ifdef __STDC__
103        double __ieee754_exp(double x)  /* default IEEE double exp */
104#else
105        double __ieee754_exp(x) /* default IEEE double exp */
106        double x;
107#endif
108{
109        double y,hi=0.0,lo=0.0,c,t;
110        int k=0.0,xsb,n0;
111        unsigned hx;
112
113        n0 = ((*(int*)&one)>>29)^1;     /* high word index */
114        hx  = *(n0+(unsigned*)&x);      /* high word of x */
115        xsb = (hx>>31)&1;               /* sign bit of x */
116        hx &= 0x7fffffff;               /* high word of |x| */
117
118    /* filter out non-finite argument */
119        if(hx >= 0x40862E42) {                  /* if |x|>=709.78... */
120            if(hx>=0x7ff00000) {
121                if(((hx&0xfffff)|*(1-n0+(int*)&x))!=0) 
122                     return x+x;                /* NaN */
123                else return (xsb==0)? x:0.0;    /* exp(+-inf)={inf,0} */
124            }
125            if(x > o_threshold) return huge*huge; /* overflow */
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  = 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            *(n0+(int*)&y) += (k<<20);  /* add k to y's exponent */
153            return y;
154        } else {
155            *(n0+(int*)&y) += ((k+1000)<<20);/* add k to y's exponent */
156            return y*twom1000;
157        }
158}
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