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29 * @(#)exp.c 8.1 (Berkeley) 6/4/93
30 * $FreeBSD: head/lib/msun/bsdsrc/b_exp.c 226414 2011-10-16 05:37:20Z das $
35 * RETURN THE EXPONENTIAL OF X
36 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
37 * CODED IN C BY K.C. NG, 1/19/85;
38 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
40 * Required system supported functions:
46 * 1. Argument Reduction: given the input x, find r and integer k such
48 * x = k*ln2 + r, |r| <= 0.5*ln2 .
49 * r will be represented as r := z+c for better accuracy.
51 * 2. Compute exp(r) by
53 * exp(r) = 1 + r + r*R1/(2-R1),
55 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
57 * 3. exp(x) = 2^k * exp(r) .
60 * exp(INF) is INF, exp(NaN) is NaN;
62 * for finite argument, only exp(0)=1 is exact.
65 * exp(x) returns the exponential of x nearly rounded. In a test run
66 * with 1,156,000 random arguments on a VAX, the maximum observed
67 * error was 0.869 ulps (units in the last place).
72 static const double p1 = 0x1.555555555553ep-3;
73 static const double p2 = -0x1.6c16c16bebd93p-9;
74 static const double p3 = 0x1.1566aaf25de2cp-14;
75 static const double p4 = -0x1.bbd41c5d26bf1p-20;
76 static const double p5 = 0x1.6376972bea4d0p-25;
77 static const double ln2hi = 0x1.62e42fee00000p-1;
78 static const double ln2lo = 0x1.a39ef35793c76p-33;
79 static const double lnhuge = 0x1.6602b15b7ecf2p9;
80 static const double lntiny = -0x1.77af8ebeae354p9;
81 static const double invln2 = 0x1.71547652b82fep0;
84 /* returns exp(r = x + c) for |c| < |x| with no overlap. */
87 __exp__D(double x, double c)
92 if (x != x) /* x is NaN */
97 /* argument reduction : x --> x - k*ln2 */
99 k = z + copysign(.5, x);
101 /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */
103 hi=(x-k*ln2hi); /* Exact. */
104 x= hi - (lo = k*ln2lo-c);
105 /* return 2^k*[1+x+x*c/(2+c)] */
107 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
110 return scalb(1.+(hi-(lo - c)), k);
112 /* end of x > lntiny */
115 /* exp(-big#) underflows to zero */
116 if(finite(x)) return(scalb(1.0,-5000));
118 /* exp(-INF) is zero */
121 /* end of x < lnhuge */
124 /* exp(INF) is INF, exp(+big#) overflows to INF */
125 return( finite(x) ? scalb(1.0,5000) : x);