1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * 4 * Use and reproduction of this software are granted in accordance with 5 * the terms and conditions specified in the Berkeley Software License 6 * Agreement (in particular, this entails acknowledgement of the programs' 7 * source, and inclusion of this notice) with the additional understanding 8 * that all recipients should regard themselves as participants in an 9 * ongoing research project and hence should feel obligated to report 10 * their experiences (good or bad) with these elementary function codes, 11 * using "sendbug 4bsd-bugs@BERKELEY", to the authors. 12 */ 13 14 #ifndef lint 15 static char sccsid[] = 16 "@(#)cosh.c 1.2 (Berkeley) 8/21/85; 1.3 (ucb.elefunt) 03/16/86"; 17 #endif not lint 18 19 /* COSH(X) 20 * RETURN THE HYPERBOLIC COSINE OF X 21 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 22 * CODED IN C BY K.C. NG, 1/8/85; 23 * REVISED BY K.C. NG on 2/8/85, 2/23/85, 3/7/85, 3/29/85, 4/16/85. 24 * 25 * Required system supported functions : 26 * copysign(x,y) 27 * scalb(x,N) 28 * 29 * Required kernel function: 30 * exp(x) 31 * exp__E(x,c) ...return exp(x+c)-1-x for |x|<0.3465 32 * 33 * Method : 34 * 1. Replace x by |x|. 35 * 2. 36 * [ exp(x) - 1 ]^2 37 * 0 <= x <= 0.3465 : cosh(x) := 1 + ------------------- 38 * 2*exp(x) 39 * 40 * exp(x) + 1/exp(x) 41 * 0.3465 <= x <= 22 : cosh(x) := ------------------- 42 * 2 43 * 22 <= x <= lnovfl : cosh(x) := exp(x)/2 44 * lnovfl <= x <= lnovfl+log(2) 45 * : cosh(x) := exp(x)/2 (avoid overflow) 46 * log(2)+lnovfl < x < INF: overflow to INF 47 * 48 * Note: .3465 is a number near one half of ln2. 49 * 50 * Special cases: 51 * cosh(x) is x if x is +INF, -INF, or NaN. 52 * only cosh(0)=1 is exact for finite x. 53 * 54 * Accuracy: 55 * cosh(x) returns the exact hyperbolic cosine of x nearly rounded. 56 * In a test run with 768,000 random arguments on a VAX, the maximum 57 * observed error was 1.23 ulps (units in the last place). 58 * 59 * Constants: 60 * The hexadecimal values are the intended ones for the following constants. 61 * The decimal values may be used, provided that the compiler will convert 62 * from decimal to binary accurately enough to produce the hexadecimal values 63 * shown. 64 */ 65 66 #ifdef VAX 67 /* static double */ 68 /* mln2hi = 8.8029691931113054792E1 , Hex 2^ 7 * .B00F33C7E22BDB */ 69 /* mln2lo = -4.9650192275318476525E-16 , Hex 2^-50 * -.8F1B60279E582A */ 70 /* lnovfl = 8.8029691931113053016E1 ; Hex 2^ 7 * .B00F33C7E22BDA */ 71 static long mln2hix[] = { 0x0f3343b0, 0x2bdbc7e2}; 72 static long mln2lox[] = { 0x1b60a70f, 0x582a279e}; 73 static long lnovflx[] = { 0x0f3343b0, 0x2bdac7e2}; 74 #define mln2hi (*(double*)mln2hix) 75 #define mln2lo (*(double*)mln2lox) 76 #define lnovfl (*(double*)lnovflx) 77 #else /* IEEE double */ 78 static double 79 mln2hi = 7.0978271289338397310E2 , /*Hex 2^ 10 * 1.62E42FEFA39EF */ 80 mln2lo = 2.3747039373786107478E-14 , /*Hex 2^-45 * 1.ABC9E3B39803F */ 81 lnovfl = 7.0978271289338397310E2 ; /*Hex 2^ 9 * 1.62E42FEFA39EF */ 82 #endif 83 84 #ifdef VAX 85 static max = 126 ; 86 #else /* IEEE double */ 87 static max = 1023 ; 88 #endif 89 90 double cosh(x) 91 double x; 92 { 93 static double half=1.0/2.0,one=1.0, small=1.0E-18; /* fl(1+small)==1 */ 94 double scalb(),copysign(),exp(),exp__E(),t; 95 96 #ifndef VAX 97 if(x!=x) return(x); /* x is NaN */ 98 #endif 99 if((x=copysign(x,one)) <= 22) 100 if(x<0.3465) 101 if(x<small) return(one+x); 102 else {t=x+exp__E(x,0.0);x=t+t; return(one+t*t/(2.0+x)); } 103 104 else /* for x lies in [0.3465,22] */ 105 { t=exp(x); return((t+one/t)*half); } 106 107 if( lnovfl <= x && x <= (lnovfl+0.7)) 108 /* for x lies in [lnovfl, lnovfl+ln2], decrease x by ln(2^(max+1)) 109 * and return 2^max*exp(x) to avoid unnecessary overflow 110 */ 111 return(scalb(exp((x-mln2hi)-mln2lo), max)); 112 113 else 114 return(exp(x)*half); /* for large x, cosh(x)=exp(x)/2 */ 115 } 116