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[] = "@(#)exp.c 4.3 (Berkeley) 08/21/85"; 16 #endif not lint 17 18 /* EXP(X) 19 * RETURN THE EXPONENTIAL OF X 20 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) 21 * CODED IN C BY K.C. NG, 1/19/85; 22 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85. 23 * 24 * Required system supported functions: 25 * scalb(x,n) 26 * copysign(x,y) 27 * finite(x) 28 * 29 * Kernel function: 30 * exp__E(x,c) 31 * 32 * Method: 33 * 1. Argument Reduction: given the input x, find r and integer k such 34 * that 35 * x = k*ln2 + r, |r| <= 0.5*ln2 . 36 * r will be represented as r := z+c for better accuracy. 37 * 38 * 2. Compute expm1(r)=exp(r)-1 by 39 * 40 * expm1(r=z+c) := z + exp__E(z,r) 41 * 42 * 3. exp(x) = 2^k * ( expm1(r) + 1 ). 43 * 44 * Special cases: 45 * exp(INF) is INF, exp(NaN) is NaN; 46 * exp(-INF)= 0; 47 * for finite argument, only exp(0)=1 is exact. 48 * 49 * Accuracy: 50 * exp(x) returns the exponential of x nearly rounded. In a test run 51 * with 1,156,000 random arguments on a VAX, the maximum observed 52 * error was .768 ulps (units in the last place). 53 * 54 * Constants: 55 * The hexadecimal values are the intended ones for the following constants. 56 * The decimal values may be used, provided that the compiler will convert 57 * from decimal to binary accurately enough to produce the hexadecimal values 58 * shown. 59 */ 60 61 #ifdef VAX /* VAX D format */ 62 /* double static */ 63 /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 64 /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */ 65 /* lnhuge = 9.4961163736712506989E1 , Hex 2^ 7 * .BDEC1DA73E9010 */ 66 /* lntiny = -9.5654310917272452386E1 , Hex 2^ 7 * -.BF4F01D72E33AF */ 67 /* invln2 = 1.4426950408889634148E0 ; Hex 2^ 1 * .B8AA3B295C17F1 */ 68 static long ln2hix[] = { 0x72174031, 0x0000f7d0}; 69 static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1}; 70 static long lnhugex[] = { 0xec1d43bd, 0x9010a73e}; 71 static long lntinyx[] = { 0x4f01c3bf, 0x33afd72e}; 72 static long invln2x[] = { 0xaa3b40b8, 0x17f1295c}; 73 #define ln2hi (*(double*)ln2hix) 74 #define ln2lo (*(double*)ln2lox) 75 #define lnhuge (*(double*)lnhugex) 76 #define lntiny (*(double*)lntinyx) 77 #define invln2 (*(double*)invln2x) 78 #else /* IEEE double */ 79 double static 80 ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 81 ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */ 82 lnhuge = 7.1602103751842355450E2 , /*Hex 2^ 9 * 1.6602B15B7ECF2 */ 83 lntiny = -7.5137154372698068983E2 , /*Hex 2^ 9 * -1.77AF8EBEAE354 */ 84 invln2 = 1.4426950408889633870E0 ; /*Hex 2^ 0 * 1.71547652B82FE */ 85 #endif 86 87 double exp(x) 88 double x; 89 { 90 double scalb(), copysign(), exp__E(), z,hi,lo,c; 91 int k,finite(); 92 93 #ifndef VAX 94 if(x!=x) return(x); /* x is NaN */ 95 #endif 96 if( x <= lnhuge ) { 97 if( x >= lntiny ) { 98 99 /* argument reduction : x --> x - k*ln2 */ 100 101 k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */ 102 103 /* express x-k*ln2 as z+c */ 104 hi=x-k*ln2hi; 105 z=hi-(lo=k*ln2lo); 106 c=(hi-z)-lo; 107 108 /* return 2^k*[expm1(x) + 1] */ 109 z += exp__E(z,c); 110 return (scalb(z+1.0,k)); 111 } 112 /* end of x > lntiny */ 113 114 else 115 /* exp(-big#) underflows to zero */ 116 if(finite(x)) return(scalb(1.0,-5000)); 117 118 /* exp(-INF) is zero */ 119 else return(0.0); 120 } 121 /* end of x < lnhuge */ 122 123 else 124 /* exp(INF) is INF, exp(+big#) overflows to INF */ 125 return( finite(x) ? scalb(1.0,5000) : x); 126 } 127