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 "@(#)exp.c 4.3 (Berkeley) 8/21/85; 1.2 (ucb.elefunt) 09/11/85"; 17 #endif not lint 18 19 /* EXP(X) 20 * RETURN THE EXPONENTIAL OF X 21 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) 22 * CODED IN C BY K.C. NG, 1/19/85; 23 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85. 24 * 25 * Required system supported functions: 26 * scalb(x,n) 27 * copysign(x,y) 28 * finite(x) 29 * 30 * Kernel function: 31 * exp__E(x,c) 32 * 33 * Method: 34 * 1. Argument Reduction: given the input x, find r and integer k such 35 * that 36 * x = k*ln2 + r, |r| <= 0.5*ln2 . 37 * r will be represented as r := z+c for better accuracy. 38 * 39 * 2. Compute expm1(r)=exp(r)-1 by 40 * 41 * expm1(r=z+c) := z + exp__E(z,r) 42 * 43 * 3. exp(x) = 2^k * ( expm1(r) + 1 ). 44 * 45 * Special cases: 46 * exp(INF) is INF, exp(NaN) is NaN; 47 * exp(-INF)= 0; 48 * for finite argument, only exp(0)=1 is exact. 49 * 50 * Accuracy: 51 * exp(x) returns the exponential of x nearly rounded. In a test run 52 * with 1,156,000 random arguments on a VAX, the maximum observed 53 * error was .768 ulps (units in the last place). 54 * 55 * Constants: 56 * The hexadecimal values are the intended ones for the following constants. 57 * The decimal values may be used, provided that the compiler will convert 58 * from decimal to binary accurately enough to produce the hexadecimal values 59 * shown. 60 */ 61 62 #ifdef VAX /* VAX D format */ 63 /* double static */ 64 /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 65 /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */ 66 /* lnhuge = 9.4961163736712506989E1 , Hex 2^ 7 * .BDEC1DA73E9010 */ 67 /* lntiny = -9.5654310917272452386E1 , Hex 2^ 7 * -.BF4F01D72E33AF */ 68 /* invln2 = 1.4426950408889634148E0 ; Hex 2^ 1 * .B8AA3B295C17F1 */ 69 static long ln2hix[] = { 0x72174031, 0x0000f7d0}; 70 static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1}; 71 static long lnhugex[] = { 0xec1d43bd, 0x9010a73e}; 72 static long lntinyx[] = { 0x4f01c3bf, 0x33afd72e}; 73 static long invln2x[] = { 0xaa3b40b8, 0x17f1295c}; 74 #define ln2hi (*(double*)ln2hix) 75 #define ln2lo (*(double*)ln2lox) 76 #define lnhuge (*(double*)lnhugex) 77 #define lntiny (*(double*)lntinyx) 78 #define invln2 (*(double*)invln2x) 79 #else /* IEEE double */ 80 double static 81 ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 82 ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */ 83 lnhuge = 7.1602103751842355450E2 , /*Hex 2^ 9 * 1.6602B15B7ECF2 */ 84 lntiny = -7.5137154372698068983E2 , /*Hex 2^ 9 * -1.77AF8EBEAE354 */ 85 invln2 = 1.4426950408889633870E0 ; /*Hex 2^ 0 * 1.71547652B82FE */ 86 #endif 87 88 double exp(x) 89 double x; 90 { 91 double scalb(), copysign(), exp__E(), z,hi,lo,c; 92 int k,finite(); 93 94 #ifndef VAX 95 if(x!=x) return(x); /* x is NaN */ 96 #endif 97 if( x <= lnhuge ) { 98 if( x >= lntiny ) { 99 100 /* argument reduction : x --> x - k*ln2 */ 101 102 k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */ 103 104 /* express x-k*ln2 as z+c */ 105 hi=x-k*ln2hi; 106 z=hi-(lo=k*ln2lo); 107 c=(hi-z)-lo; 108 109 /* return 2^k*[expm1(x) + 1] */ 110 z += exp__E(z,c); 111 return (scalb(z+1.0,k)); 112 } 113 /* end of x > lntiny */ 114 115 else 116 /* exp(-big#) underflows to zero */ 117 if(finite(x)) return(scalb(1.0,-5000)); 118 119 /* exp(-INF) is zero */ 120 else return(0.0); 121 } 122 /* end of x < lnhuge */ 123 124 else 125 /* exp(INF) is INF, exp(+big#) overflows to INF */ 126 return( finite(x) ? scalb(1.0,5000) : x); 127 } 128