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__
92 static const double
93 #else
94 static double
95 #endif
96 one = 1.0,
97 halF[2] = {0.5,-0.5,},
98 huge = 1.0e+300,
99 twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
100 o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
101 u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
102 ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
103 -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
104 ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
105 -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
106 invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
107 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
108 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
109 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
110 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
111 P5 = 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 }