2 * Copyright (c) 1985, 1993
3 * The Regents of the University of California. All rights reserved.
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33 * @(#)exp.c 8.1 (Berkeley) 6/4/93
37 * RETURN THE EXPONENTIAL OF X
38 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
39 * CODED IN C BY K.C. NG, 1/19/85;
40 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
42 * Required system supported functions:
48 * 1. Argument Reduction: given the input x, find r and integer k such
50 * x = k*ln2 + r, |r| <= 0.5*ln2 .
51 * r will be represented as r := z+c for better accuracy.
53 * 2. Compute exp(r) by
55 * exp(r) = 1 + r + r*R1/(2-R1),
57 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
59 * 3. exp(x) = 2^k * exp(r) .
62 * exp(INF) is INF, exp(NaN) is NaN;
64 * for finite argument, only exp(0)=1 is exact.
67 * exp(x) returns the exponential of x nearly rounded. In a test run
68 * with 1,156,000 random arguments on a VAX, the maximum observed
69 * error was 0.869 ulps (units in the last place).
72 * The hexadecimal values are the intended ones for the following constants.
73 * The decimal values may be used, provided that the compiler will convert
74 * from decimal to binary accurately enough to produce the hexadecimal values
80 vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000)
81 vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC)
82 vc(lnhuge, 9.4961163736712506989E1 ,ec1d,43bd,9010,a73e, 7, .BDEC1DA73E9010)
83 vc(lntiny,-9.5654310917272452386E1 ,4f01,c3bf,33af,d72e, 7,-.BF4F01D72E33AF)
84 vc(invln2, 1.4426950408889634148E0 ,aa3b,40b8,17f1,295c, 1, .B8AA3B295C17F1)
85 vc(p1, 1.6666666666666602251E-1 ,aaaa,3f2a,a9f1,aaaa, -2, .AAAAAAAAAAA9F1)
86 vc(p2, -2.7777777777015591216E-3 ,0b60,bc36,ec94,b5f5, -8,-.B60B60B5F5EC94)
87 vc(p3, 6.6137563214379341918E-5 ,b355,398a,f15f,792e, -13, .8AB355792EF15F)
88 vc(p4, -1.6533902205465250480E-6 ,ea0e,b6dd,5f84,2e93, -19,-.DDEA0E2E935F84)
89 vc(p5, 4.1381367970572387085E-8 ,bb4b,3431,2683,95f5, -24, .B1BB4B95F52683)
92 #define ln2hi vccast(ln2hi)
93 #define ln2lo vccast(ln2lo)
94 #define lnhuge vccast(lnhuge)
95 #define lntiny vccast(lntiny)
96 #define invln2 vccast(invln2)
100 #define p4 vccast(p4)
101 #define p5 vccast(p5)
104 ic(p1, 1.6666666666666601904E-1, -3, 1.555555555553E)
105 ic(p2, -2.7777777777015593384E-3, -9, -1.6C16C16BEBD93)
106 ic(p3, 6.6137563214379343612E-5, -14, 1.1566AAF25DE2C)
107 ic(p4, -1.6533902205465251539E-6, -20, -1.BBD41C5D26BF1)
108 ic(p5, 4.1381367970572384604E-8, -25, 1.6376972BEA4D0)
109 ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000)
110 ic(ln2lo, 1.9082149292705877000E-10,-33, 1.A39EF35793C76)
111 ic(lnhuge, 7.1602103751842355450E2, 9, 1.6602B15B7ECF2)
112 ic(lntiny,-7.5137154372698068983E2, 9, -1.77AF8EBEAE354)
113 ic(invln2, 1.4426950408889633870E0, 0, 1.71547652B82FE)
121 #if !defined(vax)&&!defined(tahoe)
122 if(x!=x) return(x); /* x is NaN */
123 #endif /* !defined(vax)&&!defined(tahoe) */
127 /* argument reduction : x --> x - k*ln2 */
129 k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */
131 /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */
136 /* return 2^k*[1+x+x*c/(2+c)] */
138 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
139 return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k);
142 /* end of x > lntiny */
145 /* exp(-big#) underflows to zero */
146 if(finite(x)) return(scalb(1.0,-5000));
148 /* exp(-INF) is zero */
151 /* end of x < lnhuge */
154 /* exp(INF) is INF, exp(+big#) overflows to INF */
155 return( finite(x) ? scalb(1.0,5000) : x);
158 /* returns exp(r = x + c) for |c| < |x| with no overlap. */
160 double __exp__D(x, c)
166 #if !defined(vax)&&!defined(tahoe)
167 if (x!=x) return(x); /* x is NaN */
168 #endif /* !defined(vax)&&!defined(tahoe) */
172 /* argument reduction : x --> x - k*ln2 */
174 k = z + copysign(.5, x);
176 /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */
178 hi=(x-k*ln2hi); /* Exact. */
179 x= hi - (lo = k*ln2lo-c);
180 /* return 2^k*[1+x+x*c/(2+c)] */
182 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
185 return scalb(1.+(hi-(lo - c)), k);
187 /* end of x > lntiny */
190 /* exp(-big#) underflows to zero */
191 if(finite(x)) return(scalb(1.0,-5000));
193 /* exp(-INF) is zero */
196 /* end of x < lnhuge */
199 /* exp(INF) is INF, exp(+big#) overflows to INF */
200 return( finite(x) ? scalb(1.0,5000) : x);