1/*
2 * Copyright (c) 1998, 2001, Oracle and/or its affiliates. All rights reserved.
3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
4 *
5 * This code is free software; you can redistribute it and/or modify it
6 * under the terms of the GNU General Public License version 2 only, as
7 * published by the Free Software Foundation.  Oracle designates this
8 * particular file as subject to the "Classpath" exception as provided
9 * by Oracle in the LICENSE file that accompanied this code.
10 *
11 * This code is distributed in the hope that it will be useful, but WITHOUT
12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
13 * FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
14 * version 2 for more details (a copy is included in the LICENSE file that
15 * accompanied this code).
16 *
17 * You should have received a copy of the GNU General Public License version
18 * 2 along with this work; if not, write to the Free Software Foundation,
19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
20 *
21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
22 * or visit www.oracle.com if you need additional information or have any
23 * questions.
24 */
25
26/* __ieee754_exp(x)
27 * Returns the exponential of x.
28 *
29 * Method
30 *   1. Argument reduction:
31 *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
32 *      Given x, find r and integer k such that
33 *
34 *               x = k*ln2 + r,  |r| <= 0.5*ln2.
35 *
36 *      Here r will be represented as r = hi-lo for better
37 *      accuracy.
38 *
39 *   2. Approximation of exp(r) by a special rational function on
40 *      the interval [0,0.34658]:
41 *      Write
42 *          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
43 *      We use a special Reme algorithm on [0,0.34658] to generate
44 *      a polynomial of degree 5 to approximate R. The maximum error
45 *      of this polynomial approximation is bounded by 2**-59. In
46 *      other words,
47 *          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
48 *      (where z=r*r, and the values of P1 to P5 are listed below)
49 *      and
50 *          |                  5          |     -59
51 *          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
52 *          |                             |
53 *      The computation of exp(r) thus becomes
54 *                             2*r
55 *              exp(r) = 1 + -------
56 *                            R - r
57 *                                 r*R1(r)
58 *                     = 1 + r + ----------- (for better accuracy)
59 *                                2 - R1(r)
60 *      where
61 *                               2       4             10
62 *              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
63 *
64 *   3. Scale back to obtain exp(x):
65 *      From step 1, we have
66 *         exp(x) = 2^k * exp(r)
67 *
68 * Special cases:
69 *      exp(INF) is INF, exp(NaN) is NaN;
70 *      exp(-INF) is 0, and
71 *      for finite argument, only exp(0)=1 is exact.
72 *
73 * Accuracy:
74 *      according to an error analysis, the error is always less than
75 *      1 ulp (unit in the last place).
76 *
77 * Misc. info.
78 *      For IEEE double
79 *          if x >  7.09782712893383973096e+02 then exp(x) overflow
80 *          if x < -7.45133219101941108420e+02 then exp(x) underflow
81 *
82 * Constants:
83 * The hexadecimal values are the intended ones for the following
84 * constants. The decimal values may be used, provided that the
85 * compiler will convert from decimal to binary accurately enough
86 * to produce the hexadecimal values shown.
87 */
88
89#include "fdlibm.h"
90
91#ifdef __STDC__
92static const double
93#else
94static double
95#endif
96one     = 1.0,
97halF[2] = {0.5,-0.5,},
98huge    = 1.0e+300,
99twom1000= 9.33263618503218878990e-302,     /* 2**-1000=0x01700000,0*/
100o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
101u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */
102ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */
103             -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
104ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */
105             -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
106invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
107P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
108P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
109P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
110P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
111P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
112
113
114#ifdef __STDC__
115        double __ieee754_exp(double x)  /* default IEEE double exp */
116#else
117        double __ieee754_exp(x) /* default IEEE double exp */
118        double x;
119#endif
120{
121        double y,hi=0,lo=0,c,t;
122        int k=0,xsb;
123        unsigned hx;
124
125        hx  = __HI(x);  /* high word of x */
126        xsb = (hx>>31)&1;               /* sign bit of x */
127        hx &= 0x7fffffff;               /* high word of |x| */
128
129    /* filter out non-finite argument */
130        if(hx >= 0x40862E42) {                  /* if |x|>=709.78... */
131            if(hx>=0x7ff00000) {
132                if(((hx&0xfffff)|__LO(x))!=0)
133                     return x+x;                /* NaN */
134                else return (xsb==0)? x:0.0;    /* exp(+-inf)={inf,0} */
135            }
136            if(x > o_threshold) return huge*huge; /* overflow */
137            if(x < u_threshold) return twom1000*twom1000; /* underflow */
138        }
139
140    /* argument reduction */
141        if(hx > 0x3fd62e42) {           /* if  |x| > 0.5 ln2 */
142            if(hx < 0x3FF0A2B2) {       /* and |x| < 1.5 ln2 */
143                hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
144            } else {
145                k  = invln2*x+halF[xsb];
146                t  = k;
147                hi = x - t*ln2HI[0];    /* t*ln2HI is exact here */
148                lo = t*ln2LO[0];
149            }
150            x  = hi - lo;
151        }
152        else if(hx < 0x3e300000)  {     /* when |x|<2**-28 */
153            if(huge+x>one) return one+x;/* trigger inexact */
154        }
155        else k = 0;
156
157    /* x is now in primary range */
158        t  = x*x;
159        c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
160        if(k==0)        return one-((x*c)/(c-2.0)-x);
161        else            y = one-((lo-(x*c)/(2.0-c))-hi);
162        if(k >= -1021) {
163            __HI(y) += (k<<20); /* add k to y's exponent */
164            return y;
165        } else {
166            __HI(y) += ((k+1000)<<20);/* add k to y's exponent */
167            return y*twom1000;
168        }
169}
170