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34 /* @(#)exp.c 8.1 (Berkeley) 6/4/93 */
35 #include <sys/cdefs.h>
36 __FBSDID("$FreeBSD$");
40 * RETURN THE EXPONENTIAL OF X
41 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
42 * CODED IN C BY K.C. NG, 1/19/85;
43 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
45 * Required system supported functions:
51 * 1. Argument Reduction: given the input x, find r and integer k such
53 * x = k*ln2 + r, |r| <= 0.5*ln2 .
54 * r will be represented as r := z+c for better accuracy.
56 * 2. Compute exp(r) by
58 * exp(r) = 1 + r + r*R1/(2-R1),
60 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
62 * 3. exp(x) = 2^k * exp(r) .
65 * exp(INF) is INF, exp(NaN) is NaN;
67 * for finite argument, only exp(0)=1 is exact.
70 * exp(x) returns the exponential of x nearly rounded. In a test run
71 * with 1,156,000 random arguments on a VAX, the maximum observed
72 * error was 0.869 ulps (units in the last place).
77 static const double p1 = 0x1.555555555553ep-3;
78 static const double p2 = -0x1.6c16c16bebd93p-9;
79 static const double p3 = 0x1.1566aaf25de2cp-14;
80 static const double p4 = -0x1.bbd41c5d26bf1p-20;
81 static const double p5 = 0x1.6376972bea4d0p-25;
82 static const double ln2hi = 0x1.62e42fee00000p-1;
83 static const double ln2lo = 0x1.a39ef35793c76p-33;
84 static const double lnhuge = 0x1.6602b15b7ecf2p9;
85 static const double lntiny = -0x1.77af8ebeae354p9;
86 static const double invln2 = 0x1.71547652b82fep0;
95 #if !defined(vax)&&!defined(tahoe)
96 if(x!=x) return(x); /* x is NaN */
97 #endif /* !defined(vax)&&!defined(tahoe) */
101 /* argument reduction : x --> x - k*ln2 */
103 k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */
105 /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */
110 /* return 2^k*[1+x+x*c/(2+c)] */
112 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
113 return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k);
116 /* end of x > lntiny */
119 /* exp(-big#) underflows to zero */
120 if(finite(x)) return(scalb(1.0,-5000));
122 /* exp(-INF) is zero */
125 /* end of x < lnhuge */
128 /* exp(INF) is INF, exp(+big#) overflows to INF */
129 return( finite(x) ? scalb(1.0,5000) : x);
133 /* returns exp(r = x + c) for |c| < |x| with no overlap. */
135 double __exp__D(x, c)
141 if (x != x) /* x is NaN */
146 /* argument reduction : x --> x - k*ln2 */
148 k = z + copysign(.5, x);
150 /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */
152 hi=(x-k*ln2hi); /* Exact. */
153 x= hi - (lo = k*ln2lo-c);
154 /* return 2^k*[1+x+x*c/(2+c)] */
156 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
159 return scalb(1.+(hi-(lo - c)), k);
161 /* end of x > lntiny */
164 /* exp(-big#) underflows to zero */
165 if(finite(x)) return(scalb(1.0,-5000));
167 /* exp(-INF) is zero */
170 /* end of x < lnhuge */
173 /* exp(INF) is INF, exp(+big#) overflows to INF */
174 return( finite(x) ? scalb(1.0,5000) : x);