1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms are permitted 6 * provided that the above copyright notice and this paragraph are 7 * duplicated in all such forms and that any documentation, 8 * advertising materials, and other materials related to such 9 * distribution and use acknowledge that the software was developed 10 * by the University of California, Berkeley. The name of the 11 * University may not be used to endorse or promote products derived 12 * from this software without specific prior written permission. 13 * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR 14 * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED 15 * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE. 16 * 17 * All recipients should regard themselves as participants in an ongoing 18 * research project and hence should feel obligated to report their 19 * experiences (good or bad) with these elementary function codes, using 20 * the sendbug(8) program, to the authors. 21 */ 22 23 #ifndef lint 24 static char sccsid[] = "@(#)cosh.c 5.3 (Berkeley) 06/30/88"; 25 #endif /* not lint */ 26 27 /* COSH(X) 28 * RETURN THE HYPERBOLIC COSINE OF X 29 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 30 * CODED IN C BY K.C. NG, 1/8/85; 31 * REVISED BY K.C. NG on 2/8/85, 2/23/85, 3/7/85, 3/29/85, 4/16/85. 32 * 33 * Required system supported functions : 34 * copysign(x,y) 35 * scalb(x,N) 36 * 37 * Required kernel function: 38 * exp(x) 39 * exp__E(x,c) ...return exp(x+c)-1-x for |x|<0.3465 40 * 41 * Method : 42 * 1. Replace x by |x|. 43 * 2. 44 * [ exp(x) - 1 ]^2 45 * 0 <= x <= 0.3465 : cosh(x) := 1 + ------------------- 46 * 2*exp(x) 47 * 48 * exp(x) + 1/exp(x) 49 * 0.3465 <= x <= 22 : cosh(x) := ------------------- 50 * 2 51 * 22 <= x <= lnovfl : cosh(x) := exp(x)/2 52 * lnovfl <= x <= lnovfl+log(2) 53 * : cosh(x) := exp(x)/2 (avoid overflow) 54 * log(2)+lnovfl < x < INF: overflow to INF 55 * 56 * Note: .3465 is a number near one half of ln2. 57 * 58 * Special cases: 59 * cosh(x) is x if x is +INF, -INF, or NaN. 60 * only cosh(0)=1 is exact for finite x. 61 * 62 * Accuracy: 63 * cosh(x) returns the exact hyperbolic cosine of x nearly rounded. 64 * In a test run with 768,000 random arguments on a VAX, the maximum 65 * observed error was 1.23 ulps (units in the last place). 66 * 67 * Constants: 68 * The hexadecimal values are the intended ones for the following constants. 69 * The decimal values may be used, provided that the compiler will convert 70 * from decimal to binary accurately enough to produce the hexadecimal values 71 * shown. 72 */ 73 74 #if defined(vax)||defined(tahoe) 75 #ifdef vax 76 #define _0x(A,B) 0x/**/A/**/B 77 #else /* vax */ 78 #define _0x(A,B) 0x/**/B/**/A 79 #endif /* vax */ 80 /* static double */ 81 /* mln2hi = 8.8029691931113054792E1 , Hex 2^ 7 * .B00F33C7E22BDB */ 82 /* mln2lo = -4.9650192275318476525E-16 , Hex 2^-50 * -.8F1B60279E582A */ 83 /* lnovfl = 8.8029691931113053016E1 ; Hex 2^ 7 * .B00F33C7E22BDA */ 84 static long mln2hix[] = { _0x(0f33,43b0), _0x(2bdb,c7e2)}; 85 static long mln2lox[] = { _0x(1b60,a70f), _0x(582a,279e)}; 86 static long lnovflx[] = { _0x(0f33,43b0), _0x(2bda,c7e2)}; 87 #define mln2hi (*(double*)mln2hix) 88 #define mln2lo (*(double*)mln2lox) 89 #define lnovfl (*(double*)lnovflx) 90 #else /* defined(vax)||defined(tahoe) */ 91 static double 92 mln2hi = 7.0978271289338397310E2 , /*Hex 2^ 10 * 1.62E42FEFA39EF */ 93 mln2lo = 2.3747039373786107478E-14 , /*Hex 2^-45 * 1.ABC9E3B39803F */ 94 lnovfl = 7.0978271289338397310E2 ; /*Hex 2^ 9 * 1.62E42FEFA39EF */ 95 #endif /* defined(vax)||defined(tahoe) */ 96 97 #if defined(vax)||defined(tahoe) 98 static max = 126 ; 99 #else /* defined(vax)||defined(tahoe) */ 100 static max = 1023 ; 101 #endif /* defined(vax)||defined(tahoe) */ 102 103 double cosh(x) 104 double x; 105 { 106 static double half=1.0/2.0,one=1.0, small=1.0E-18; /* fl(1+small)==1 */ 107 double scalb(),copysign(),exp(),exp__E(),t; 108 109 #if !defined(vax)&&!defined(tahoe) 110 if(x!=x) return(x); /* x is NaN */ 111 #endif /* !defined(vax)&&!defined(tahoe) */ 112 if((x=copysign(x,one)) <= 22) 113 if(x<0.3465) 114 if(x<small) return(one+x); 115 else {t=x+exp__E(x,0.0);x=t+t; return(one+t*t/(2.0+x)); } 116 117 else /* for x lies in [0.3465,22] */ 118 { t=exp(x); return((t+one/t)*half); } 119 120 if( lnovfl <= x && x <= (lnovfl+0.7)) 121 /* for x lies in [lnovfl, lnovfl+ln2], decrease x by ln(2^(max+1)) 122 * and return 2^max*exp(x) to avoid unnecessary overflow 123 */ 124 return(scalb(exp((x-mln2hi)-mln2lo), max)); 125 126 else 127 return(exp(x)*half); /* for large x, cosh(x)=exp(x)/2 */ 128 } 129