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36 /* @(#)exp.c 8.1 (Berkeley) 6/4/93 */
37 #include <sys/cdefs.h>
38 __FBSDID("$FreeBSD$");
42 * RETURN THE EXPONENTIAL OF X
43 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
44 * CODED IN C BY K.C. NG, 1/19/85;
45 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
47 * Required system supported functions:
53 * 1. Argument Reduction: given the input x, find r and integer k such
55 * x = k*ln2 + r, |r| <= 0.5*ln2 .
56 * r will be represented as r := z+c for better accuracy.
58 * 2. Compute exp(r) by
60 * exp(r) = 1 + r + r*R1/(2-R1),
62 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
64 * 3. exp(x) = 2^k * exp(r) .
67 * exp(INF) is INF, exp(NaN) is NaN;
69 * for finite argument, only exp(0)=1 is exact.
72 * exp(x) returns the exponential of x nearly rounded. In a test run
73 * with 1,156,000 random arguments on a VAX, the maximum observed
74 * error was 0.869 ulps (units in the last place).
79 static const double p1 = 0x1.555555555553ep-3;
80 static const double p2 = -0x1.6c16c16bebd93p-9;
81 static const double p3 = 0x1.1566aaf25de2cp-14;
82 static const double p4 = -0x1.bbd41c5d26bf1p-20;
83 static const double p5 = 0x1.6376972bea4d0p-25;
84 static const double ln2hi = 0x1.62e42fee00000p-1;
85 static const double ln2lo = 0x1.a39ef35793c76p-33;
86 static const double lnhuge = 0x1.6602b15b7ecf2p9;
87 static const double lntiny = -0x1.77af8ebeae354p9;
88 static const double invln2 = 0x1.71547652b82fep0;
97 #if !defined(vax)&&!defined(tahoe)
98 if(x!=x) return(x); /* x is NaN */
99 #endif /* !defined(vax)&&!defined(tahoe) */
103 /* argument reduction : x --> x - k*ln2 */
105 k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */
107 /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */
112 /* return 2^k*[1+x+x*c/(2+c)] */
114 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
115 return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k);
118 /* end of x > lntiny */
121 /* exp(-big#) underflows to zero */
122 if(finite(x)) return(scalb(1.0,-5000));
124 /* exp(-INF) is zero */
127 /* end of x < lnhuge */
130 /* exp(INF) is INF, exp(+big#) overflows to INF */
131 return( finite(x) ? scalb(1.0,5000) : x);
135 /* returns exp(r = x + c) for |c| < |x| with no overlap. */
137 double __exp__D(x, c)
143 if (x != x) /* x is NaN */
148 /* argument reduction : x --> x - k*ln2 */
150 k = z + copysign(.5, x);
152 /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */
154 hi=(x-k*ln2hi); /* Exact. */
155 x= hi - (lo = k*ln2lo-c);
156 /* return 2^k*[1+x+x*c/(2+c)] */
158 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
161 return scalb(1.+(hi-(lo - c)), k);
163 /* end of x > lntiny */
166 /* exp(-big#) underflows to zero */
167 if(finite(x)) return(scalb(1.0,-5000));
169 /* exp(-INF) is zero */
172 /* end of x < lnhuge */
175 /* exp(INF) is INF, exp(+big#) overflows to INF */
176 return( finite(x) ? scalb(1.0,5000) : x);