2 * Copyright (c) 1985, 1993
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33 * @(#)exp__E.c 8.1 (Berkeley) 6/4/93
37 * ASSUMPTION: c << x SO THAT fl(x+c)=x.
38 * (c is the correction term for x)
41 * / exp(x+c) - 1 - x , 1E-19 < |x| < .3465736
45 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
46 * KERNEL FUNCTION OF EXP, EXPM1, POW FUNCTIONS
47 * CODED IN C BY K.C. NG, 1/31/85;
48 * REVISED BY K.C. NG on 3/16/85, 4/16/85.
50 * Required system supported function:
54 * 1. Rational approximation. Let r=x+c.
57 * exp(r) - 1 = ---------------------- ,
58 * cosh(r/2) - sinh(r/2)
59 * exp__E(r) is computed using
60 * x*x (x/2)*W - ( Q - ( 2*P + x*P ) )
61 * --- + (c + x*[---------------------------------- + c ])
63 * where P := p1*x^2 + p2*x^4,
64 * Q := q1*x^2 + q2*x^4 (for 56 bits precision, add q3*x^6)
67 * (See the listing below for the values of p1,p2,q1,q2,q3. The poly-
68 * nomials P and Q may be regarded as the approximations to sinh
70 * sinh(r/2) = r/2 + r * P , cosh(r/2) = 1 + Q . )
72 * The coefficients were obtained by a special Remez algorithm.
74 * Approximation error:
76 * | exp(x) - 1 | 2**(-57), (IEEE double)
77 * | ------------ - (exp__E(x,0)+x)/x | <=
78 * | x | 2**(-69). (VAX D)
81 * The hexadecimal values are the intended ones for the following constants.
82 * The decimal values may be used, provided that the compiler will convert
83 * from decimal to binary accurately enough to produce the hexadecimal values
89 vc(p1, 1.5150724356786683059E-2 ,3abe,3d78,066a,67e1, -6, .F83ABE67E1066A)
90 vc(p2, 6.3112487873718332688E-5 ,5b42,3984,0173,48cd, -13, .845B4248CD0173)
91 vc(q1, 1.1363478204690669916E-1 ,b95a,3ee8,ec45,44a2, -3, .E8B95A44A2EC45)
92 vc(q2, 1.2624568129896839182E-3 ,7905,3ba5,f5e7,72e4, -9, .A5790572E4F5E7)
93 vc(q3, 1.5021856115869022674E-6 ,9eb4,36c9,c395,604a, -19, .C99EB4604AC395)
95 ic(p1, 1.3887401997267371720E-2, -7, 1.C70FF8B3CC2CF)
96 ic(p2, 3.3044019718331897649E-5, -15, 1.15317DF4526C4)
97 ic(q1, 1.1110813732786649355E-1, -4, 1.C719538248597)
98 ic(q2, 9.9176615021572857300E-4, -10, 1.03FC4CB8C98E8)
101 #define p1 vccast(p1)
102 #define p2 vccast(p2)
103 #define q1 vccast(q1)
104 #define q2 vccast(q2)
105 #define q3 vccast(q3)
111 const static double zero=0.0, one=1.0, half=1.0/2.0, small=1.0E-19;
112 double z,p,q,xp,xh,w;
113 if(copysign(x,one)>small) {
116 #if defined(vax)||defined(tahoe)
117 q = z*( q1 +z*( q2 +z* q3 ));
118 #else /* defined(vax)||defined(tahoe) */
120 #endif /* defined(vax)||defined(tahoe) */
125 c += x*((xh*w-(q-(p+xp)))/(one-w)+c);
128 /* end of |x| > small */
131 if(x!=zero) one+small; /* raise the inexact flag */
132 return(copysign(zero,x));