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e_exp.c (5189B)
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1 /* @(#)e_exp.c 5.1 93/09/24 */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
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 #ifndef lint
14 static char rcsid[] = "$FreeBSD: src/lib/msun/src/e_exp.c,v 1.9 2003/07/23 04:53:46 peter Exp $";
15 #endif
16
17 /* __ieee754_exp(x)
18 * Returns the exponential of x.
19 *
20 * Method
21 * 1. Argument reduction:
22 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
23 * Given x, find r and integer k such that
24 *
25 * x = k*ln2 + r, |r| <= 0.5*ln2.
26 *
27 * Here r will be represented as r = hi-lo for better
28 * accuracy.
29 *
30 * 2. Approximation of exp(r) by a special rational function on
31 * the interval [0,0.34658]:
32 * Write
33 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
34 * We use a special Reme algorithm on [0,0.34658] to generate
35 * a polynomial of degree 5 to approximate R. The maximum error
36 * of this polynomial approximation is bounded by 2**-59. In
37 * other words,
38 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
39 * (where z=r*r, and the values of P1 to P5 are listed below)
40 * and
41 * | 5 | -59
42 * | 2.0+P1*z+...+P5*z - R(z) | <= 2
43 * | |
44 * The computation of exp(r) thus becomes
45 * 2*r
46 * exp(r) = 1 + -------
47 * R - r
48 * r*R1(r)
49 * = 1 + r + ----------- (for better accuracy)
50 * 2 - R1(r)
51 * where
52 * 2 4 10
53 * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
54 *
55 * 3. Scale back to obtain exp(x):
56 * From step 1, we have
57 * exp(x) = 2^k * exp(r)
58 *
59 * Special cases:
60 * exp(INF) is INF, exp(NaN) is NaN;
61 * exp(-INF) is 0, and
62 * for finite argument, only exp(0)=1 is exact.
63 *
64 * Accuracy:
65 * according to an error analysis, the error is always less than
66 * 1 ulp (unit in the last place).
67 *
68 * Misc. info.
69 * For IEEE double
70 * if x > 7.09782712893383973096e+02 then exp(x) overflow
71 * if x < -7.45133219101941108420e+02 then exp(x) underflow
72 *
73 * Constants:
74 * The hexadecimal values are the intended ones for the following
75 * constants. The decimal values may be used, provided that the
76 * compiler will convert from decimal to binary accurately enough
77 * to produce the hexadecimal values shown.
78 */
79
80 #include "math.h"
81 #include "math_private.h"
82
83 static const double
84 one = 1.0,
85 halF[2] = {0.5,-0.5,},
86 huge = 1.0e+300,
87 twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
88 o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
89 u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
90 ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
91 -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
92 ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
93 -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
94 invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
95 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
96 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
97 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
98 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
99 P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
100
101
102 double
103 __ieee754_exp(double x) /* default IEEE double exp */
104 {
105 double y,hi=0.0,lo=0.0,c,t;
106 int32_t k=0,xsb;
107 u_int32_t hx;
108
109 GET_HIGH_WORD(hx,x);
110 xsb = (hx>>31)&1; /* sign bit of x */
111 hx &= 0x7fffffff; /* high word of |x| */
112
113 /* filter out non-finite argument */
114 if(hx >= 0x40862E42) { /* if |x|>=709.78... */
115 if(hx>=0x7ff00000) {
116 u_int32_t lx;
117 GET_LOW_WORD(lx,x);
118 if(((hx&0xfffff)|lx)!=0)
119 return x+x; /* NaN */
120 else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
121 }
122 if(x > o_threshold) return huge*huge; /* overflow */
123 if(x < u_threshold) return twom1000*twom1000; /* underflow */
124 }
125
126 /* argument reduction */
127 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
128 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
129 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
130 } else {
131 k = invln2*x+halF[xsb];
132 t = k;
133 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
134 lo = t*ln2LO[0];
135 }
136 x = hi - lo;
137 }
138 else if(hx < 0x3e300000) { /* when |x|<2**-28 */
139 if(huge+x>one) return one+x;/* trigger inexact */
140 }
141 else k = 0;
142
143 /* x is now in primary range */
144 t = x*x;
145 c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
146 if(k==0) return one-((x*c)/(c-2.0)-x);
147 else y = one-((lo-(x*c)/(2.0-c))-hi);
148 if(k >= -1021) {
149 u_int32_t hy;
150 GET_HIGH_WORD(hy,y);
151 SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */
152 return y;
153 } else {
154 u_int32_t hy;
155 GET_HIGH_WORD(hy,y);
156 SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */
157 return y*twom1000;
158 }
159 }