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33 * @(#)acosh.c 8.1 (Berkeley) 6/4/93
37 * RETURN THE INVERSE HYPERBOLIC COSINE OF X
38 * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS)
39 * CODED IN C BY K.C. NG, 2/16/85;
40 * REVISED BY K.C. NG on 3/6/85, 3/24/85, 4/16/85, 8/17/85.
42 * Required system supported functions :
45 * Required kernel function:
46 * log1p(x) ...return log(1+x)
50 * acosh(x) = log [ x + sqrt(x*x-1) ]
52 * acosh(x) := log1p(x)+ln2, if (x > 1.0E20); else
53 * acosh(x) := log1p( sqrt(x-1) * (sqrt(x-1) + sqrt(x+1)) ) .
54 * These formulae avoid the over/underflow complication.
57 * acosh(x) is NaN with signal if x<1.
58 * acosh(NaN) is NaN without signal.
61 * acosh(x) returns the exact inverse hyperbolic cosine of x nearly
62 * rounded. In a test run with 512,000 random arguments on a VAX, the
63 * maximum observed error was 3.30 ulps (units of the last place) at
64 * x=1.0070493753568216 .
67 * The hexadecimal values are the intended ones for the following constants.
68 * The decimal values may be used, provided that the compiler will convert
69 * from decimal to binary accurately enough to produce the hexadecimal values
75 vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000)
76 vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC)
78 ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000)
79 ic(ln2lo, 1.9082149292705877000E-10,-33, 1.A39EF35793C76)
82 #define ln2hi vccast(ln2hi)
83 #define ln2lo vccast(ln2lo)
89 double t,big=1.E20; /* big+1==big */
91 #if !defined(vax)&&!defined(tahoe)
92 if(x!=x) return(x); /* x is NaN */
93 #endif /* !defined(vax)&&!defined(tahoe) */
95 /* return log1p(x) + log(2) if x is large */
96 if(x>big) {t=log1p(x)+ln2lo; return(t+ln2hi);}
99 return(log1p(t*(t+sqrt(x+1.0))));