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1 :
2 : /* @(#)e_exp.c 1.6 04/04/22 */
3 : /*
4 : * ====================================================
5 : * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
6 : *
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 : //#include <sys/cdefs.h>
14 : //__FBSDID("$FreeBSD$");
15 :
16 : /* __ieee754_exp(x)
17 : * Returns the exponential of x.
18 : *
19 : * Method
20 : * 1. Argument reduction:
21 : * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
22 : * Given x, find r and integer k such that
23 : *
24 : * x = k*ln2 + r, |r| <= 0.5*ln2.
25 : *
26 : * Here r will be represented as r = hi-lo for better
27 : * accuracy.
28 : *
29 : * 2. Approximation of exp(r) by a special rational function on
30 : * the interval [0,0.34658]:
31 : * Write
32 : * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
33 : * We use a special Remes algorithm on [0,0.34658] to generate
34 : * a polynomial of degree 5 to approximate R. The maximum error
35 : * of this polynomial approximation is bounded by 2**-59. In
36 : * other words,
37 : * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
38 : * (where z=r*r, and the values of P1 to P5 are listed below)
39 : * and
40 : * | 5 | -59
41 : * | 2.0+P1*z+...+P5*z - R(z) | <= 2
42 : * | |
43 : * The computation of exp(r) thus becomes
44 : * 2*r
45 : * exp(r) = 1 + -------
46 : * R - r
47 : * r*R1(r)
48 : * = 1 + r + ----------- (for better accuracy)
49 : * 2 - R1(r)
50 : * where
51 : * 2 4 10
52 : * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
53 : *
54 : * 3. Scale back to obtain exp(x):
55 : * From step 1, we have
56 : * exp(x) = 2^k * exp(r)
57 : *
58 : * Special cases:
59 : * exp(INF) is INF, exp(NaN) is NaN;
60 : * exp(-INF) is 0, and
61 : * for finite argument, only exp(0)=1 is exact.
62 : *
63 : * Accuracy:
64 : * according to an error analysis, the error is always less than
65 : * 1 ulp (unit in the last place).
66 : *
67 : * Misc. info.
68 : * For IEEE double
69 : * if x > 7.09782712893383973096e+02 then exp(x) overflow
70 : * if x < -7.45133219101941108420e+02 then exp(x) underflow
71 : *
72 : * Constants:
73 : * The hexadecimal values are the intended ones for the following
74 : * constants. The decimal values may be used, provided that the
75 : * compiler will convert from decimal to binary accurately enough
76 : * to produce the hexadecimal values shown.
77 : */
78 :
79 : #include <float.h>
80 :
81 : #include "math_private.h"
82 :
83 : static const double
84 : one = 1.0,
85 : halF[2] = {0.5,-0.5,},
86 : o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
87 : u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
88 : ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
89 : -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
90 : ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
91 : -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
92 : invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
93 : P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
94 : P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
95 : P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
96 : P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
97 : P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
98 :
99 : static volatile double
100 : huge = 1.0e+300,
101 : twom1000= 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0*/
102 :
103 : double
104 0 : __ieee754_exp(double x) /* default IEEE double exp */
105 : {
106 0 : double y,hi=0.0,lo=0.0,c,t,twopk;
107 0 : int32_t k=0,xsb;
108 : u_int32_t hx;
109 :
110 0 : GET_HIGH_WORD(hx,x);
111 0 : xsb = (hx>>31)&1; /* sign bit of x */
112 0 : hx &= 0x7fffffff; /* high word of |x| */
113 :
114 : /* filter out non-finite argument */
115 0 : if(hx >= 0x40862E42) { /* if |x|>=709.78... */
116 0 : if(hx>=0x7ff00000) {
117 : u_int32_t lx;
118 0 : GET_LOW_WORD(lx,x);
119 0 : if(((hx&0xfffff)|lx)!=0)
120 0 : return x+x; /* NaN */
121 0 : else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
122 : }
123 0 : if(x > o_threshold) return huge*huge; /* overflow */
124 0 : if(x < u_threshold) return twom1000*twom1000; /* underflow */
125 : }
126 :
127 : /* argument reduction */
128 0 : if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
129 0 : if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
130 0 : hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
131 : } else {
132 0 : k = (int)(invln2*x+halF[xsb]);
133 0 : t = k;
134 0 : hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
135 0 : lo = t*ln2LO[0];
136 : }
137 0 : STRICT_ASSIGN(double, x, hi - lo);
138 : }
139 0 : else if(hx < 0x3e300000) { /* when |x|<2**-28 */
140 0 : if(huge+x>one) return one+x;/* trigger inexact */
141 : }
142 0 : else k = 0;
143 :
144 : /* x is now in primary range */
145 0 : t = x*x;
146 0 : if(k >= -1021)
147 0 : INSERT_WORDS(twopk,0x3ff00000+(k<<20), 0);
148 : else
149 0 : INSERT_WORDS(twopk,0x3ff00000+((k+1000)<<20), 0);
150 0 : c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
151 0 : if(k==0) return one-((x*c)/(c-2.0)-x);
152 0 : else y = one-((lo-(x*c)/(2.0-c))-hi);
153 0 : if(k >= -1021) {
154 0 : if (k==1024) {
155 0 : double const_0x1p1023 = pow(2, 1023);
156 0 : return y*2.0*const_0x1p1023;
157 : }
158 0 : return y*twopk;
159 : } else {
160 0 : return y*twopk*twom1000;
161 : }
162 : }
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