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32 /* @(#)exp.c 8.1 (Berkeley) 6/4/93 */
33 #include <sys/cdefs.h>
34 __FBSDID("$FreeBSD$");
38 * RETURN THE EXPONENTIAL OF X
39 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
40 * CODED IN C BY K.C. NG, 1/19/85;
41 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
43 * Required system supported functions:
49 * 1. Argument Reduction: given the input x, find r and integer k such
51 * x = k*ln2 + r, |r| <= 0.5*ln2 .
52 * r will be represented as r := z+c for better accuracy.
54 * 2. Compute exp(r) by
56 * exp(r) = 1 + r + r*R1/(2-R1),
58 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
60 * 3. exp(x) = 2^k * exp(r) .
63 * exp(INF) is INF, exp(NaN) is NaN;
65 * for finite argument, only exp(0)=1 is exact.
68 * exp(x) returns the exponential of x nearly rounded. In a test run
69 * with 1,156,000 random arguments on a VAX, the maximum observed
70 * error was 0.869 ulps (units in the last place).
75 static const double p1 = 0x1.555555555553ep-3;
76 static const double p2 = -0x1.6c16c16bebd93p-9;
77 static const double p3 = 0x1.1566aaf25de2cp-14;
78 static const double p4 = -0x1.bbd41c5d26bf1p-20;
79 static const double p5 = 0x1.6376972bea4d0p-25;
80 static const double ln2hi = 0x1.62e42fee00000p-1;
81 static const double ln2lo = 0x1.a39ef35793c76p-33;
82 static const double lnhuge = 0x1.6602b15b7ecf2p9;
83 static const double lntiny = -0x1.77af8ebeae354p9;
84 static const double invln2 = 0x1.71547652b82fep0;
93 #if !defined(vax)&&!defined(tahoe)
94 if(x!=x) return(x); /* x is NaN */
95 #endif /* !defined(vax)&&!defined(tahoe) */
99 /* argument reduction : x --> x - k*ln2 */
101 k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */
103 /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */
108 /* return 2^k*[1+x+x*c/(2+c)] */
110 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
111 return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k);
114 /* end of x > lntiny */
117 /* exp(-big#) underflows to zero */
118 if(finite(x)) return(scalb(1.0,-5000));
120 /* exp(-INF) is zero */
123 /* end of x < lnhuge */
126 /* exp(INF) is INF, exp(+big#) overflows to INF */
127 return( finite(x) ? scalb(1.0,5000) : x);
131 /* returns exp(r = x + c) for |c| < |x| with no overlap. */
133 double __exp__D(x, c)
139 if (x != x) /* x is NaN */
144 /* argument reduction : x --> x - k*ln2 */
146 k = z + copysign(.5, x);
148 /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */
150 hi=(x-k*ln2hi); /* Exact. */
151 x= hi - (lo = k*ln2lo-c);
152 /* return 2^k*[1+x+x*c/(2+c)] */
154 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
157 return scalb(1.+(hi-(lo - c)), k);
159 /* end of x > lntiny */
162 /* exp(-big#) underflows to zero */
163 if(finite(x)) return(scalb(1.0,-5000));
165 /* exp(-INF) is zero */
168 /* end of x < lnhuge */
171 /* exp(INF) is INF, exp(+big#) overflows to INF */
172 return( finite(x) ? scalb(1.0,5000) : x);